Flat modules and rings finitely generated as modules ... - Springer Link

Mathematical Notes, Vol. 60, No. s 1996

Flat M o d u l e s a n d R i n g s F i n i t e l y G e n e r a t e d as M o d u l e s over T h e i r C e n t e r A. A. T u g a n b a e v

UDC 512.55

ABSTRACT. A module is called distributive (is said to be a chain module) if the lattice o f all its submodules is distributive (is a chain). Let a ring A be a finitely generated module over its unitary central subring R. We prove the equivalence of the following conditions: (1) A is a right or left distributive semiprime ring; (2) for any maximal ideal M of a subring R central in A, the ring of quotients AM is a finite direct product of semihereditary Bdzout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields; (3) all right ideals and all left ideals of the ring A are flat (right and left) modules over the ring A, and A is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields. KEY W O R D S : distributive ring, chain ring, semiprime ring, Bdzout domain, Jacobson radical, flat module, prime ideal, nilpotent element, completely closed module.

All rings are assumed to be associative and with a nonzero identity element; the modules are assumed to be unitary. Expressions such as % chain ring" mean that the corresponding right and left conditions hold. A module is called distributive (is said to be a chain module) if the lattice of all submodules is distributive (is a chain). A module for which all finitely generated submodules are cyclic is called a B~zo~tt module. For a commutative ring A, the following conditions are equivalent [1]: (1) all ideals of the ring A are flat modules over A; (2) A is a distributive semiprime ring; (3) all localizations of the ring A by maximal ideals are chain domains. The main result of the paper is Theorem 1, which follows from Propositions 1.14 and 2.13. T h e o r e m 1. Let a ring A be a finitely generated module over its unitary centrad subring R. Then the following conditions are equivalent: (1) A is a right or left distributive semiprime ring; (2) for any max./mal idea/ M of the subring R, which is central in A, the ring of quotients AM is a finite direct product of semihereditary Bdzout domains whose quotient rings by their Jaeobson radicals are finite direct products of skew fields; (3) ail submodtdes of fiat right modules and fiat leR modules over the ring A are flat modules, and A is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideais are semihereditary orders in skew fields. In connection with Theorem 1, we note that the ring of matrices over a field gives an example of a ring A that is a finitely generated module over its center and is not right distributive or left distributive, while all A-modules are fiat. Moreover, there exists a distributive semiprime ring A that contains a principal right ideal aA that is not a fiat right module over A (see 2.15). The proof of Theorem 1 is decomposed into several assertions. Let us present the necessary notation and definitions. An element of the ring is said to be regular if its fight annihilator r(a) coincides with its left annihilator e(a) and is trivial. For a module M we denote by End(M), J ( M ) , m a x ( M ) , Lat(M), Sing(M), and sg(M) the ring of endomorphisms, the Jacobson radical, the set of all maximal submodules, the lattice of all submodules, the singular submodule of the module M , and the ideal of the ring End(M) formed by all endomorphisms with essential kernels, respectively. For a ring A we denote by C(A) and Translated from Matematicheskie Zametl~, Vol. 60, No. 2, pp. 254-277, August, 1996. Original article submitted May 16, 1995. 186

0001-4346/96/6012-0186 $12.50

~)1997 Plenum Publishing Corporation

U(A) the center and the group of invertible elements of A, respectively. Let F be a subset of the module MA over a ring A and let G E Lat(M). Denote by ( F : G) the right ideal {a E A [ Fa C_ G} of the ring A, which is an ideal whenever F E Lat(M). If ( F : fit) is an ideal of the ring A, then ( F : G) = ((FA) : G), where FA is the submodule of the module M generated by its subset F . A module MA is said to be multiplicative if the following equivalent conditions hold: (1) for any N E Lat(M), there exists an ideal B of the ring A such that N = M B ; (2) N = M ( M : N) for every N E Lat(M). A module M is said to be antisingular (singular) if Sing(M) = 0 (Sing(M) = M). A module is said to be uniform if any two its nonzero submodules have nonzero intersection. A module is said to be (Goldie) finite-dimensional if it contains no infinite direct sums of nonzero submodules. By a subquotient we mean a submodule of a quotient module. A module is said to be semi-Noetherian (semi-Artin) if each of its nonzero subquotient modules has a simple quotient module (submodule). A submodule N of a module M is said to be closed if each submodule N1 9 Lat(M) that is an essential extension of the module N coincides with N . An endomorphism f of the module M is said to be locally nilpotent if for any element m 9 M there exists a positive integer n = n(m) such that f " ( m ) = 0. A ring with a distributive lattice of two-sided ideals is called an arithmetic ring. A module is said to be invariant (quasi-invariant) if all its submodules (respectively, all maximal submodules) are completely invariant submodules of the original module. A ring A is right quasi-invariant (right invariant) iff all maximal right ideals (respectively, all right ideals) of the ring A are ideals iff all cyclic right A-modules are quasi-invariant (respectively, invariant). A ring A is right quasi-invariant iff A / J ( A ) is right quasi-invariant. A prime ideal of the ring A not containing other prime ideals of the ring A is said to be a minimal prime ideal. A proper ideal is said to be completely prime if the quotient ring by this ideal is a domain, i.e., contains no zero divisors. A ring is said to be normal if all idempotents are central. A ring without nonzero nilpotent dements is called a reduced ring. A subset T of a ring A is said to be: (1) mulfiplicafive if T is closed under the multiplication, contains the identity element, and does not contain the zero element of the ring A; (2) righf commutafive if for any two elements a 9 A and t 9 T , there exist elements b 9 A and u 9 T such that au = tb; (3) right reversive if for any elements a 9 A and t 9 T such that ta = 0, there exists an element u E T such that au = O. A subset T of a ring A is called a set of right denominators, provided that the following two equivalent conditions [2~ p. 648 of the Russian translation I are satisfied: (1) T is a right commutative right reversive multiplicative subset of A; (2) there exists a ring AT and a ring homomorphism fT -- f: A --~ AT such that

f ( T ) C_ U(AT),

AT = { f ( a ) f ( t ) - ' [a 9 A, t 9 T } ,

and Ker(f) = {a E A l at = 0 for some t E T}. Under these conditions, the ring AT is called the right ring of quotients of the ring A with respect to T and fT is called the canonical homomorphism, and for any element a E A we write aT instead of fT(a). For each right ideal B of the ring A we denote by BT the right ideal of the ring AT generated by the set fT(B). If T = A \ M , where M is a right ideal of the ring A, then we write fM, AM, aM, and BM instead of f r , AT, aT, and BT, respectively. A ring A is said to be right localizable if for each maximal right ideal M of it, the right ring of quotients AM exists. The left ring of quotients TA with respect to a set of left denominators T and the canonical homomo~hism T f : A ~ TA are defined similarly. If T = A \ M , where M is a left ideal of the ring A, then we also write M instead of the subscript T. Left localizable rings are defined similarly to right localizable ones.

w 1.1. L e m m a .

B~zout modules and distributive m o d u l e s

For a module MA over a ring A, the following conditions are equiva/ent: 187

(1) M is a distributive module; (2) all 2-generated submodules of the module M are distributive; (3) all subquotients of the module M a r e distributive; (4) all subquotients of the module M are quasiinvariant; (5) M has no 2-generated submodules H with quotient modules of the form F ~ G, where F and G are isomorphic simple modules; (6) for any elements rn, n 9 M , ~.here exist elements a, b E A such that 1 = a + b and m a A + nbA C m A M nA ;

(7) all 2-generated semisimple subquotients of the module M are quasiinvarian~.; (8) for an arbitrary subquotient M of the module M and for any elements m , n G M such that ~ A M ~ A = O, there exJst elements a, b G A such that 1 = a + b and ~ a = ~b = 0 ; (9) A = (m : nA) + (n : m A ) for any elements rn, n G A; (10) (rn + n)A = m A N ( m + n)A + n A N ( m + n)A for any elements m , n 9 M ; (11) for any elements m , n 9 M , there exists a right ideal B of the ring A such that (m + n)A = mB+ nB; (12) f ( H ) C_ H for any closed submodule H of an arbitrary subquotient N of the module M and any homomorphism f : H ---} N ; (13) Horn(F, G) = 0 for any submodules F and G of an arbitrary subquot.ient N of the module M such that F n G = 0; (14) f ( H ) C_ H for any simple subquotient H of an arbitrary subquotient N of the module M and any homomorphism f : H ---, N ; (15) f ( H ) C_ H for any semi-Artin submodule H of an arbitrary subquotient N of the module M and any homomorphism f : H ~ N ; (16) any subquotient of the module M each of whose subdirectly indecomposable subquotients is semiNoetherian, is an invariant module. P r o o f . The equivalence of conditions (1)-(6), as well as the implications (1) :. (12) and (1) ~- (13), was proved in [3]. The equivalence of conditions (6), (8), and (9), as well as the equivalence of conditions (5) and (7), and the equivalence of conditions (2) and (10), and also the implications (12) ~ (7), (16) '.. (7), (13) ---} (14), (14) :. (15), and (15) ---4- (7) can be verified directly. The implication (11) .~ (10) follows from the fact that if (rn + n)A = m B + n B , then by applying the module law we obtain the relations m A N (rn + n)A = m A N (rob + nB) = m B

and

n A N (m + n)A = n A N (rob + n B ) = nB.

The implication (9) --->. (10) follows from the fact that for m, n e M , f --- m + n , and B - (m : fA), we have B = (n : f A ) , m A f3 f A = roB, and n A f t f A = riB. Let us prove the implication (4) ;. (16). Since condition (4) holds for all subquotients of the module M , we restrict ourselves to the proof of the fact that the module M is invariant. Let M ~ N G Lat(M), let N C_ H E Lat(M), and let M / H be a subdirectly indecomposable module. Let us prove that H is a completely invariant submodule of M . It suffices to prove that H coincides with the intersection F of all completely invariant submodules of the module M that contain H . Assume the contrary. Since, by assumption, the module M / H is semi-Noetherian, it follows that a nonzero submodule F / H has a maximal submodule G / H . Then G is a maximal submodule of the module F that contains the module H . Since G is completely invariant in F and F is a completely invariant submodule of M , it follows that G is a completely invariant submodule of the module M that is strictly contained in F and contains the module H , which contradicts the choice of the. module F . Since M / N is the subdirect product of subdirectly indecomposable modules and any intersection of completely invariant submodules of the module M is completely invariant in M , we see that N is completely invariant in M . Q

188

1.2. If all 2-generated submodules MA are multiplicative, then M is a distributive module. Indeed, let m , n E M , fi =_ m + n, and N -- m A + n A . By assumption, there exists an ideal B of the ring A such that f A = N B . Since B is an ideal, we have

N B = (rnA + n A ) B = rnB + nB. By 1.1, M is distributive. 1.3. L e m m a . For a module M A over a right invariant ring A, the following conditions are equivalent: (1) M is a distributive module; (2) A = ( F : G) + (G : F) for any two cyclic submodules F and G of the module M ; (3) A = ( F : G) + (G : F ) for any finitely generated submodules F and G of the module M ; (4) for any finitely generated submodules F and G of the module M , there exist dements a, b E A such that l = a + b and Fa + Gb C_ F n G. P r o o f . The equivalence of (3) and (4), as well as the implication (3) ==~ (2), can be verified directly. The equivalence of (1) and (2) follows from 1.1 (see condition (9)) and from the fact that for the case of a right invariant ring A, for any elements rn, n E M we have (m : nA) -- (rnA : nA). Let us prove the implication (2) .~ (3). We shall use the following directly verified assertion: ( . ) if X , Y I , . . . , Y, are ideals of the ring A and A = X + Y/ for all j , then A = X + I"1.--Y, =

x + ni=l Yi. Let

~rt

n

i=I

j=l

By assumption (2), we have A = ( f i A : giA) + (giA: fiA) for all i and j . Therefore,

A=(fiA:C)+(gjA:

fiA)

Vi,j.

It follows from ( . ) that n

A = ( f i A : G) + ~ ( g i A :

fiA) = ( f i A : C) + ( G : fiA)Vi.

j=l

Therefore, A = ( f i A : G) + (G: F) for all i. Then it follows from ( , ) that

A = n(s,A :G)+(G:F)=(F:G)+(G:F).

0

i=l

1.4. P r o p o s i t i o n . For a module MA over a right invariant ring A, the following conditions are equivalent: (1) M is a distributive module; (2) all finitely generated submodules of the module M are maltiplicative; (3) for any finitely generated submodule N of the module M and for an arbitrary Bnitely generated submodule F of the module N , there exists a finitely generated ideal B of the ring A such that F - N B ; (4) all 2-generated submodules of the module M are multiplicative. P r o o f . The implications (3) '.- (4) and (2) '., (4) are obvious, and the implication (4) ~ (1) follows from 1.2. (1) '.. (2). Let n be a positive integer, let N be an arbitrary n-generated submodule of the module M , and let T E L a t ( N ) . We must prove the existence of an ideal B of the ring A such that T = N B . We perform induction on n. For n = 1, the assertion holds because any cyclic module over a right invariant ring is a multiplicative module (since this module is isomorphic to a quotient module of a free cyclic 189

module). Let the and G is a cyclic are ideals D and have A = ( f : G)

assertion hold for n < k a n d N = F + G , where the module F has k - 1 g e n e r a t o r s module. By the induction assumption, F t3 G = F ( F : G) = G(G : F ) ; moreover, there E of the ring A such that T CI F = F D and T t3 G = GE. Furthermore, by 1.3 we + ( G : F). Then

F = F ( F : G) + F(G:

F) =

a ( a : F) + F ( G : F) = N ( G : F).

The relation G = N ( F : G) can be proved similarly. We denote by B the ideal (G : F)D + (F : G)E of the ring A. Therefore,

T = Tf3 F + Tf3 a = F D + GE = N(G : F)D + N ( F : G)E = N B , and B is the desired ideal. (2) ~ (3). Let m

lti

N= ZxiA

and

F= Zf

i=1

j.

j=l

Condition (2) implies the existence of elements bij E (N : F ) , 1 < i < m , 1 _< j _< n, such that m

fj = ~

zibijYj.

i=l

Then the ideal B of the ring A generated by all elements bit is the desired finitely generated ideal.

[]

1.5. l'f MA is a Noetherian module over a right invariant ring A, then M is distributive if and only if all submodules of the module M are multiplicative. (This follows from 1.4.) 1.6. Let MA be a module over a right distributive right invariant ring A and let B and D be right ideals of the ring A. Then M B f3 M D = M ( B N D). Indeed, let s

t

i=I

j=l

where rni, nj E M , bi E B , and dj E D. Let $

t

and

F=EbiAC_A

G-Ed

j CA.

j=l

i=l

Since F and G are finitely generated right ideals of the right distributive right invariant ring A, it follows from 1.3 (see condition (4)) that there exist elements a, b E A such that 1 = a + b and bin, djb E F t3 G for all i, j . Then s

x = x. +

t

b,a + i=1

a,b

M ( F n O) c

M(B riD).

j=l

Therefore, M B f3 M D C_ M ( B N D). The converse inclusion is obvious. 1.7. Let MA be a finitely generated distributive module over a right invariant ring A. Then M is an invariant module. Indeed, let f E End(M) and N E Lat(M). By 1.4 we have N = M B , where B is an ideal of the ring A. Then f ( N ) = / ( M B ) C_/(M)B C_M B = N . 1.8. Let A be a ring with a polynomial identity. Then we have the following assertions: (1) every prime quotient ring B of the ring A is an order in a simple Artin ring, and each of essential right and left ideals of the ring B contains a nonzero central element; (2) every semiprime quotient ring of the ring A is antisingular; (3) if A is a domain, then the ring A is uniform. (Assertions (1) and (2) follow from [4, 6.1.30, 6.1.48, 6.28, 6.31], and (3) follows from (1).) 190

1.9. Let a ring A be a finitely generated module over a unitary subring R of the center of A. Then have the following assertions: (1) for any M E max(R), the ring of quotients AM is a finitely generated module over its local central subring RM ; (2) if N is an ideal of the ring A that is finitely generated as a right or as a left idea/, then N is a finitely generated module over a central subring of A, and hence is finitely generated both as a right and as a left ideal of the ring A; (3) AJCR) C_ JCA); (4) AMJ(RM) C J(AM) for any M e max(R); (5) for any M e max(R), the ring of quotients AM is semilocal; (6) any prime quotient ring of the ring A is an order in a simple Artin ring; (7) if A is a right chain right Artin ring, then A is a chain Artin ring. Assertions (1) and (2) can be verified directly. Assertion (3) is proved in [5, Chap. 3, Corollary 0.5]. Assertion (4) follows from (3) applied to AM and RM. Assertion (5) follows from (4) and from the fact that (AM)/(AMJ(RM)) is a finite-dimensional algebra over the field RM/J(RM). Assertion (6) follows from 1.8 (1). Let us prove (7). Let N be a nonzero radical of the ring A, let F be a field that is the center of the skew field Q - A / N , let n be the dimension of the vector space Q over F , and let T - N / N 2 . Since TA ~ QA, the dimension of the vector space T over F is equal to n. Then AT ~- AQ. Therefore, the principal right ideal N is also a principal left ideal, and hence A is a chain Artin ring. we

1.10. Let max(R) be the set of all maximal ideals of the central subring R of the ring A and let F and G be submodules of the module Aa (for example, the intersection, the sum, or the product of one-sided ideals of the ring A). Then we have the following assertions: (1) if FM = GM for all M e max(R), then F = G; (2) (FG)M = FMGM, (F + G)M = FM + GM, and (F M G)M = FM M GM for all M e max(R); (3) if M E max(R) and T is an arbitrary right (left) ideal of the ring AM, then there exists a right (left) ideal B of the ring A such that BM = T; (4) an element a E A is zero if and only if the canonical image of this element in the ring of quotients AM is zero for any M e max(R); (5) the ring A is reduced (semiprime) if and only if any ring AM is reduced (semiprime) for any M e max(R); (6) the ring A is right (left) distributive if and only if the ring AM is right (left) distributive for any

M e max(R). Assertions (t), (2), (3), and (4) are well known and can be veriaed directly, assertion (5) foUows from (4) and (t), and (8) follows from (t) and (3) and from the fact that by (2) we have ((B + D) N E)M = (BM + DM) C1EM

and

(B N E + D N E)M = BM N EM + DM N EM.

1.11 [3]. A is a right distributive semilocal ring iff A is a right B6zout ring and A / J ( A ) is the finite direct product of skew fields. 1.12. Let a ring A be a finitely generated module over a unitary central subring R. Then A is right distributive iff for any M E max(R), the ring of quotients AM is a right B6zout ring whose quotient ring by the Jacobson radical is the finite direct product of skew fields. (This follows from 1.9 (5), 1.10 (6) and 1.11.) 1.13 [3]. A is a semilocal reduced right B6zout ring iff A is the finite direct product of right semihereditary right uniform semilocal right B6zout domains over which the right cyclic modules are finitedimensional. 1.14. P r o p o s i t i o n . Let a ring A be a finitely generated module over a unitary central subring R. Then the following conditions are equiva/ent: (1) A is a right distributive semiprime ring; (2) A is a left distributive semiprime ring; (3) for any M 6 max(R), the ring of quotients AM is the finite direct product of semihereditary uniform Bdzout domains whose quotient rings by their Jaeobson radicals are finite direct products of skew fields. 191

Proof. Since (3) is symmetric, it sumces to prove that (1)iff(3). The implication (3) ~- (1) follows from 1.10 (5) and 1.12. (1) ~ (3). By 1.8 (2) and 1.22 (3), A is a reduced ring. Let M E max(R) and B _-- A u . By 1.10 (5) and 1.12, B is a reduced right B~zout ring and A / J ( A ) is the finite direct product of skew fields. Now we apply 1.13.

1.15 [3]. Let Q be the right ring of quotients of a ring A with respect to a set of right denominators T, Q = AT, and let f : A --* Q be the canonical homomorphism. Then we have the following assertions: (1) for any elements q l , . . . , q, E Q there are elements t E T and a l , . . . , a,, E A such that ql - f ( a i ) f ( t ) - l for i = 1 , . . . , n ; (2) if B is a right ideal of the ring A, then BT = { f ( b ) f ( t ) -1 [b E B , t E T}; (3) if B is a right ideal of the ring A, a E A, and f(a) E BT, then at E B for some element t E T; (4) ( B + D )T = BT + DT , ( B N D )T = BT n DT for any right ideals B, D of the ring A; (5) if N is a right ideal of the ring Q and E = f - ~ (N N f ( A ) ) , then E is a right ideal of the ring A and N - ET ; (6) if G is a finitely generated submodule of the left module I(A)Q, then there exists an element t E T such that Gf(t) C_ f ( A ) , and hence the left module I(A)G is isomorphic to the finitely generated ideal

aS(t) of the ring f(A); (7) if A is a left distributive ring, then the rings Q and f ( A ) are left distributive and I(A)Q is a distributive left module, and the ring f ( A ) contains all idempotents and all nilpotent elements of the

gQ; (8) if A is a right distributive ring, then Q is a right distributive ring as well; (9) if Q is a right distributive ring, then for any right ideals B , D, and E of the ring A, we have the relation (B N ( D + E))T = (B N D + BOlE)T; (10) if the ring A / K e r ( f ) is right finite-dimensional (right uniform, right Noetherian, right Artin, right semi-Noetherian, right semi-Artin), then Q is a right finite-dimensional (right uniform, right Noetherian, right Artin, right semi-Noetherian, right semi-Artin) ring; (11) if T is also a set of left denominators in the ring A, then TA ~ Q, and the kernels of the canonical homomorphisms A -~ Q and A --* TA coincide; (12) ff T = A \ M , where M is a right ideal of the ring A, then Q is a local ring and J(Q) = MT =--MM. 1.16. L e m m a . Let A be a right localizable ring. For any M E max(AA) we denote by H ( M ) the kernel of the canonical homomorphism A ~ AM. Then: (1) if a is an element of the ring A, B is a right ideal of the ring A, and a M E B M for alJ M E maX(AA), then a E B; (2) if a E A and aM = 0 f o r all M E max(AA), then a = 0; (3) if B and D are right ideals of the ring A and BM = DM for all M E maX(AA), then B = D; (4) the ring A is reduced if and only if for any M E max(AA), the ring AM is reduced; (5) if B is a right ideal of the ring A such that BM is an ideal of the ring AM for any M E max(AA), then B is an ideal of the ring A; (6) if for any M E max(AA), the ring AM is right invariant, then A is a right invariant ring; (7) every maximal right ideal of the ring A is a completely prime ideal of the ring A and A / M is a skew field; (8) the ring A is right distributive if and only if AM is a right chain ring for any M E max(AA); (9) if A is right distributive, M E max(AA), and the quotient ring A / H ( M ) is right semi-Noetherian, then AM is a right semi-Noetherian right invariant right chain ring; (10) if A is right distributive, M E max(Aa), and the quotient ring A / H ( M ) is a right semi-Artin ring, then AM is right Artin right invariant right chain ring; (11) if A is right distributive and for any M E max(Aa), the quotient ring A / H ( M ) is right semiNoetherian or a right semi-Artin ring, then A is a right invariant ring. P r o o f . Items (1)-(8) are proved in [31. (9), (10). Let Q - AM. By item (8) and Lemma 1.15 (10), G is a right chain ring that is right semi-Noetherian or a right semi-Artin ring. By Lemma 1.1 (see conditions (15) and (16)), Q is a right 192

invariant ring. Now assume that A / H ( M ) is a right semi-Artin ring. By item (6), A is right invariant. By Lemma 1.15 (10), Q is a right chain right semi-Artin ring. Then Q is left perfect and satisfies the minimality condition for prindpal right ideals. Therefore, QQ is a chain module with the minimality condition, and hence an Artin module. (11). The assertion follows from items (6), (9), and (10). ['7 1.17. L e m m a . (1) A is a right distributive semiprime ring whose quotient ring by any minimal prime ideal is right antisingular iff A is a right distributive right antisingular ring iff A is a right distributive reduced ring iff for any maximal ideal M of the ring A, the right ring of quotients AM ex/sts and is a right chain domain; (2) if A is a right distributive reduced ring whose quotient ring by any minimal prime ideal is left finite-dimensional, then A is a left distributive ring; (3) if A is a right distributive ring over which all subdirectly indecomposable cyclic right modules are semi-Noetherian, then A is right invariant; (4) if A is a right distributive right Noetherian left finite-dimensional prime ring, then A is an invariant ch'stributive Noetherian domain; (5) if A is a right distributive semiprime ring whose quotient ring by any minimal prime ideal is right semi-Noetherian, then A is a reduced ring such that for any M E max(AA), the right ring of quotients AM exists and is a right chain right invariant semi-Noetherian domain; (6) if A is a right distributive semiprime ring whose quotient ring by any minimal prime ideal is a right Noetherian lef~ finite-dimensional ring, then A is an invariant distributive ring whose any prime quotient ring is an invariant Noetherian domain, and for any maximal ideal M of A, the rings AM and MA exist, are/somorphic, and are invariant chain domains of principal ideals.

Proof. Items (i) and (2) are proved in [3]. (3) followsfrom I.I (see condition (16). (4). Since the prime ring A is right invariantby item (3), A is a domain. For domains, item (3) is proved in [3]. (5). Let F(A) be the set of all right semi-Noetherian quotient rings of A by minimal prime ideals and let B E F(A). By item (4), B is a right invariant prime ring, and hence a domain. Then A is a subdirect product of domains, and hence a reduced ring. By item (1), for any M e max(AA), the ring AM exists and is a right chain domain, and therefore the kernel H of the canonical homomorphism A ~ AM is a completely prime ideal, and hence A / H is a homomorphic image of a right semi-Noetherian ring from F(A). Then A / H is a right semi-Noetherian ring, and we can apply Lemma 1.15 (9), (11). (6). By items prime ideal is an distributive ring. and are invariant

(4) and (5), A is a right invariant reduced ring whose quotient ring over any minimal invariant Noetherian domain. By the symmetric analog of item (5), A is an invariant By item (5) and by Lemma 1.15 (10), (11), the rings AM and MA exist, are isomorphic, chain Noetherian domains. Then AM and MA are domains of principal ideals. []

1.18 [3]. Any right B6zout module over a right quasiinvariant ring is a distributive module. If A is a right B$zout ring, then A is right distributive if and only if A is right quasiinvariant. 1.19. A module MA over a strictly regular ring A is distributive if and only if M is a B6zout module. Sufficiency follows from 1.18. Let us prove necessity: It suffices to show that the module N = rnA+nA C_M is cyclic. By 1.1, there are a, b E A such that 1 = a + b, ma E hA, and nb E m A . There exists a central idempotent t of the strictly regular ring A such that aA = tA. Then we have

rnt e ntA

and

n(1 - t) = (ha + nb)(1 - t) = nb(1 - t) e m(1 - t)A.

Therefore, N = N~ 6) N ( I - ~) = m~A + B~A + m ( l - t)A + n ( l - ~)A = (nt + m ( l - ~))A.

193

1.20. MA is a distributive module iff any 2-generated semisimple quotient module of an arbitrary 2generated submodule T of the module M is distributive iff for any 2-generated submodule T of the module MA, the module T / T J ( A ) over the ring A/J(A) is distributive. (This follows from 1.1 (see condition (6)) and from the fact that the lattices Lat((N/NJ(A))A/j(A)) and L a t ( ( N / N J ( A ) ) A ) coincide.)

be strictly regular (for example, this is the case if A/ J( A ) is the direct product of skew fields). The module M is distributive if and only if M is a Bdzout module. 1.21.

P r o p o s i t i o n . Let MA be a module over the ring A, and let the quotient ring A / J ( A )

In Proposition 1.21, sufficiency follows from 1.18, and necessity can be verified by means of 1.19, 1.20, and the Nakayama lemma. Q 1.22. C o r o l l a r y . A is a right distributive ring and the quotient ring A / J ( A ) is regular itf A is a right Bdzout ring and the quotient ring A~ J( A) is strictly regular. Since strictly regular rings coincide with right distributive regular rings, it follows that Corollary 1.22 is a consequence of 1.21. [] 1.23. A is a right distributive right antisingular semilocai ring iff A is a right antisingular right B~zout ring, and A / J ( A ) is the finite direct product of skew fields iff A is a finite direct product of right uniform right B$zout Ai domains and for any i, Ai/J(Ai) is the finite direct product of skew fields, and all cyclic right modules over A are finite-dimensional. (1.23 follows from 1.11, 1.13, and 1.17.) 1.24. (1) If all simple subquotients of the module M are isomorphic, then M is distributive iff M is a chain module. (2) If A is a local ring, then MA is distributive iff M is a B~zout module iff M is a chain module. (For (1), see [3]. Assertion (2) follows from 1.21 and (1).) w

Flat modules

We need a series of well-known assertions on module over the ring A. Then for any left ideal the canonical group homomorphism fB : E | AB of the ring A that contains B and i : B --* M is

tensor products and fiat modules. Let EA B of the ring A, by the rule f B ( ~ ej | bj) ~ E B is well defined. If, in this case, M is the natural embedding, then fB = fM(lv, |

be a right = ~ ejbj, a left ideal

i).

2.1. Let EA and AM be a right module and a left module over a ring A. Then we have the following assertions: (1) if B and C are submodules of the module AM, then the intersection in E @(B + C) of the canonical images of the groups E @ B and E @ C coincides with the canonical image of the group E | (B gl C) ; (2) if A M = ~ t e T Amt and if {et}teT is a set of elements E with at most finitely many nonzero elements, then the relation ~ t e T et | m t = 0 holds if and only if there exist a finite set J , a finite set { z j } j e l of elements of the module E , and a set {ait}ieJ, tET of elements of the ring A with at most finitely many nonzero elements such that

Z ajtrnt = 0Vj tET

6 J

and

el = E xjajtVt E T. jEJ

(2.1 follows from [6, Chap. 1, w Lemmas 7 and 10].) 2.2. A module EA over the ring A is called a Hettory torsion-free module if for an arbitrary elements e 6 E and a 6 A such that ea = 0, there are elements fi 6 E and bi 6 A, where i = 1, . . . , n, such that e = ~f~=l f~bi and bia = 0 for all i. (1) EA is a Hattory torsion-free module iff any cyclic submodule of M is contained in a Hattory torsion-free module that belongs to M iff for any element a 6 A, the natural group homomorphism E @ Aa ---, Ea is an isomorphism; (2) any flat module is a Hattory torsion-free module; (3) if any cyclic submodule of the module M is contained in a flat module that belongs to M , then M is a Hattory torsion-free module. The first equivalence of (1) can be verified directly, and the second equivalence is verified by means of 2.1 (2). Assertion (2) follows from (1) and [2, Proposition 11.20]. Assertion (3) follows from (2) and (1). 194

2.3. L e m m a . For a module E.4 over a ring A , the following conditions are equivalent: (1) E is a flat module; (2) for any 6nitely generated left ideal M of the ring A , the canon/cal group homomorphism E @ M E M is an isomorphism; (3) the canonical group homomorphism E @ M --. E M is an isomorphism for any left ideal M of the ring A; (4) if E ~- F / G , where F is a free module, then for any element g E G, there exists a homomorphism h: F --. G such that h(g ) = g ; (5) i f E ~ F / G , where F is a free module, then for any t~nlte set gi of elements of the module G, there is a homomorphism h : F --, G such that h(gi) = gi for all gi ; (6) any tinitely generated submodu/e of the module E is contained in a fiat submodule of the module E; (7) E B N E C = E ( B f3 C) for any left ideals B and C of the ring A, and for any element a E A , the natural group homomorphism E @ Aa --* Ea is an isomorphism; (8) E B N E C = E ( B N C) for any tinitely generated left ideals B and C of the ring A, and for any element a 6 A , the natural group homomorphism E @ Aa ~ Ea is an isomorphism; (9) EA is a Hattory torsion-free module, and E B N E C = E ( B f3 C) for any finitely generated left ideals B and C of the ring A; (10) any cyclic submodule of the module M is contained in a fiat submodule o f the module M , and M B I3 M C = M ( B f3 C) for any ~qnitely generated left ideals B and C of the ring A .

Proof. The equivalence of assertions (1), (2), (3), (4), and (5) is proved in [2, Propositions 11.20 and 11.27], the implication (6) ~- (1) is proved in [7, Corollary 10.4.6]~ and the implications (I) ~ (6) and (7) ~- (8) are evident. The implication (3) ~. (7) follows from 2.1 (1). The equivalence of conditions (8) and (9) follows from 2.2. The implication (10) ==~ (9) follows from 2.2 (3). The implication (1) ~ (10) foUows from the equi,~aJence of conditions (1) and (3). Let us prove the implication (8) '.. (2). By induction on the positive integer t, we shall prove that for any left ideal M of the ring A generated by t elements, the canonical group homomorphism fM : E | M ~ E M is an isomorphism. For t = 1, the assertion holds by assumption (8). Assume that the assertion holds for t < k and let us prove it for t = k. Let us represent the left ideal M in the form M = B + C , where B is a (k - 1)-generated left ideal and C is a principal left ideal. Let f s : E@B---* E B , hB: E |

fv: E|

E|

hc: E |

~ EC,

w: E | A ---* E A ,

E|

g: E |

E|

be the canonical group homomorphisms. Let = ~

e, | (b, - c,) = Z

~, | b, - ~

~, | c, ~ Xer(SM),

b, ~ B,

c, ~ C,

z = ~

e,c, ~ ~ B ,

and let y be an element of E | such that ~ e i | = A s ( y ) . Since 0 = f M ( x ) = ~ e i ( b i - r follows that z = ~ eie~ E E B r E C . By assumption, E B M E C = E ( B N C). Then z = ~ u j d i , where uj E E and dj E B M C . Let v be an element from E | B such that ~ u j | dj = h s ( v ) . Then 0 = SM(~

uj | d~ - ~

e, ~ b , ) = wg(h~(~ - y)) = S~(~ -

~),

and hence v = y, because fB is a monomorphism by the induction assumption. Then

The relation ~ ei | ci = ~ u i | d i can be proved similarly. Therefore,

= ~ Then f is an isomorphism.

~ | b~ - ~

~i | ci = O.

[] 195

2.4. For a ring A, the following conditions are equivalent: (1) all principal right ideals are flat (right modules over A); (2) all principal left ideals are flat (left modules over A); (3) for any elements m, n E A such that mn --- 0, there exist elements a, b E A such that a % b = 1, m a = 0, and h a = 0 . Since (3) is symmetric, it suffices to prove the equivalence of (1) and (3). Let m E A. Since m A A A / r ( m ) , and we can identify H o m ( A A , r ( m ) ) with r ( m ) , it follows from 2.3 (see condition (4)) that { m A is a flat module} iff {for any element n E r(rn) there exists an element a E r ( m ) such that n = an } iff {if rnn = O, then there exist elements a, b E A such that a + b = 1, m a = O, and bn = 0 }. Therefore, (1) and (3) are equivalent. 2.5. (1) All submodules of the module MA are flat iff all cyclic submodules of the module M are flat and N B M N C = N ( B N C) for any finitely generated submodule N of the module M and any finitely generated left ideals B and C of the ring A. (2) All submodules of flat right modules and flat left modules over a ring A are flat iff all finitely generated right ideals of the ring A are flat iff all finitely generated left ideals of the ring A are flat iff for any finitely generated right ideal E and any finitely generated left ideal M of the ring A, the natural group homomorphism E | M ~ E M is an isomorphism iff all right ideals and all left ideals of the ring A are flat. Assertion (1) follows from 2.3. In (2), the first and the second equivalences are proved in [8, 39.12], the third equivalence follows from 2.3, and the fourth one follows from the previous equivalences. 2.6. Let all principal left ideals of the ring A be flat, and let E be a right ideal of the ring A. The following conditions are equivalent: (1) EA is a flat module; (2) E B N E C = E ( B N C) for any finitely generated left ideals B and C of the ring A; (3) EMBM N EMCM = EM(BM N CM) for any finitely generated left ideals B and C of the ring A and for any maximal ideal M of a subring R of the center of the ring A. The equivalence of (1) and (2) follows from 2.3 (see conditions (3) and (7)), and the equivalence of conditions (2) and (3) follows from 1.10. 2.7. P r o p o s i t i o n . For a reduced ring A , the following conditions are equivalent: (1) all principal right ideals o f the ring A are fiat; (2) for all elements m , n E A such that (rn)N(n) = 0, there are elements a, b E A such (m) n = 0, and (b) n (n) = 0; (3) for all elements rn, n e A such that (rn) N (n) = O, there are elements a, b E A such m a = O, and nb = 0; (4) m A @ n A = ( m + n ) A for all elements m, n e A such that (m) N (n) = 0; (5) for all elements m , n e A such that (m) M (n) = O, the 2-generated right ideal principal right ideal; (6) (rn + n ) A = (rn + n ) A n m A ~ (rn + n ) A N n A for all elements m , n E A such that

that a + b = 1, that a + b = 1,

m A fl~ n A is a (rn) N (n) = O.

P r o o f . The equivalence of conditions (1), (2), and (3) follows from the Lemmas 1.5 (1) and 2.4. The implications (4) ~- (5) and (4) ~ (6) are obvious. (3) '.- (4). It suffices to prove the relations m E ( r n § and n E ( m + n ) A . By assumption (3), there are elements a, b E A such that a + b = 1, m a = O, and nb = O. Then ( m + n)b = m b = m a + r o b = m and ( m + n)a = na = na + nb = n. (5) ' ~- (3). By assumption, there exist elements f , g, u, v e A such that m = ( m J: + ng)u and n = ( r n f + ng)v. Since (rn) M (n) = 0, it follows that rn = r n f u and 0 = r n f v = n(1 - gv) = nb, where b = 1 - gv. Let a = 1 - b = gv e A . Since v E r ( m f ) , it follows from Lemma 1.5 (1) that ugv e r ( m f ) and rna = m f u a = r n f u g v = O. (6) ~ (4). If Y ~ ( m + n ) A and g l = m A @ n A , and if h,n: g l ~ m A and h,~: N1 ~ m A are the natural projections, then it follows from condition (6) that Y = h m ( g ) @ h n ( N ) . Therefore, m = h , , ( m + n) e N , n = h,~(rn + n) e N , and N = N1. [:]

196

2.8. If A is a right distributive ring, then A is fight antisingular iff A is a reduced ring and all its principal right ideals are fiat. (Assertion 2.8 follows from 2.7 and 1.22 (3).) 2.9. All submodules of flat modules over a reduced fight Bdzout ring are fiat. (Assertion 2.9 follows from 2.7 and 2.5 (2).) 2.10. P r o p o s i t i o n . For a ring A , the following conditions are equivalent: (1) all submodules of fiat (right and left) modules over the ring A are fiat; (2) E B N E C = E ( B f3 C) for any finitely generated right ideal E and for all finitely generated left ideals B and C of the ring A , and for all dements m, n E A such that m n = 0, there are elements a, b E A such that a + b = l , ma = O , and bn = O; (3) E M B M N EMCM = E M ( B M N CM) for all finitely generated left ideals B and C of the ring A and any max/ran/ideM M of a unitary subring R of the center of the ring A , and all principal right ideMs of the ring A are nat. Assertion 2.10 follows from 2.5 (2) and 2.6, because by 2.4, without loss of generality, we may assume that all principal left ideals of the ring A are fiat. 2.11. Let all principal right ideals of a ring A be fiat and let R be a unitary subring of C ( A ) . Then the following conditions are equivalent: (1) if E is a right ideal of the ring A and EM is a fiat right over the ring AM for any M E max(R), then E A is a f l a t module; (2) if for any M E max(R), all submodules of fiat modules over the ring AM are fiat, then all submodules of flat modules over the ring A are fiat; (3) if for any M E max(R), the ring AM is right or left semihereditary, then all submodules of fiat modules over the ring A are fiat; (4) if for any M 6 max(R), the ring AM is a reduced right or left B4zout ring, then all submodules of flat modules of the ring A are fiat. In 2.11, item (1) follows from 2.3 and 2.6, (2) follows from 2.3 and 2.10, (3) follows from (2) and 2.5 (2), and (4) follows from (2) and 2.9. 2.12. Let A be a unitary subring of a classically semisimple ring B. Then the following assertions hold: (1) any finitely generated flat (right or left) module over A is projective; (2) if all finitely generated right ideals of the ring A are fiat, then A is a semihereditary ring. Since all modules over a classically semisimple ring are projective, assertion (1) follows from the following well-known assertion [9, Chap. 1, Proposition 11.6]: if M is a finitely generated flat right module over a unitary subring A of a ring B and if M | A B is a projective right module over the ring B , then MA is a projective module. Since it follows from 2.5 under the conditions of item (2) that both right ideals and left ideals of the ring A are flat, we see that assertion (2) follows from (1). 2.13. P r o p o s i t i o n . Let a right of left distributive ring A be a finitely generated module over its central unitary subring R . Then we have the following assertions: (1) A is a distributive reduced ring over which al.1submodules of flat right modules and fiat left modules are flat; (2) any prime quotient ring of the ring A is a semihereditary order in a skew t~eld. P r o o f . (1). By Proposition 1.14, A is a distributive ring, and for any M E max(R), the ring of quotients AM is a semihereditary reduced ring. By 1.10 (5), A is a reduced ring. By 2.8, all principal right ideals of the ring A are fiat. Since all rings AM are semihereditary, it follows from 2.11 (3) and 2.5 that all submodules of fiat modules over the ring A are fiat. Let us prove (2). Let B be a prime quotient ring of the ring A. By 1.8 (1), B is an order in a simple Artin ring Q. By (1) applied to B we see that B is a distributive reduced ring all of whose right ideals are flat. By 2.12, B is a semihereditary ring. Since Q is a simple Artin classical ring of quotients of a reduced ring, Q is a skew field. [] 197

2.14. Let A be a local ring all of whose principal right ideals are flat, and let m and n be elements of the ring A such that m n = 0. By 2.4, there are elements a, b G A such that a + b = 1, rna = O, and bn = 0. Since the ring A is local, at least one of the elements a and b is invertible. Therefore, at least one of the elements m and n is zero. Then A is a domain. 2.15. Let A be a chain prime ring that is not a domain (an example of such a ring is given in [11]). By 2.14, not all principal right ideals of the ring A are flat. 2.16. Let M A be a module over a distributive invariant ring A. The module M is flat iff any cyclic submodule of the module M is contained in a flat submodule of the module M iff M is a Hattory torsion-free module. (This follows from 1.6 and 2.3.)

2.17. L e m m a . Let al , a2 , bl , and b2 be elements o f a ring A such that albl = a2b2 and al +a2 ~ B , where B is a fiat right ideal. Then there are elements fii E A , 1 < i, j < 2, such that al.fll = a2.f21, Proof. BA be an Since d E such that

( 1 - /11)bl = f12b2,

and

alfl2 =a2f~2,

( 1 - f22)b2 = f~lb1.

Let DA be a free module of rank 2 with basis {dl, d2}, let d = dlbl + d2b2 E D , let g: DA ---) epimorphism such that g(dlyl + d2Y2) = alyl - a2y2 for all yl, y2 E A, and let N - Ker(g). N and B is a flat module, it follows from 2.3 that there exists a h o m o m o r p h i s m f : D ~ N f ( d ) = d. Let f i j , where 1 < i, j _< 2, be elements of the ring A such that f ( d l ) = d l f l l + d2f21 E N

and

f(d~) = d i l l 2 + d2f22 E N.

Then

O-~-g(f(dl)):g(f(d2)),

alfll

=a2f21,alfl2

----"a2f22,

d = f ( d ) = f ( d l "t- d2b2) -- d1(fllbl -1" f1262) -F d2(/21 bl + f2262), bl -- f11bl + f12b2,

b2 : f21bl + fz2b2,

(1 - fxl)bl -- f12b2,

(1 - f22)b2 = f21b1.

[]

2.18. M o d u l e s o f q u o t i e n t s . Suppose T is the set of right denominators in a ring A, f : A A T -- Q is a canonical ring homomorphism, and NA is a module over A. By a module of quotients of the module N with respect to the set T we mean the right Q-module N | Q, which is denoted by N T . Denote by gT the A-module homomorphism gT -- g: N --, N T such that g(n) = n | 1. If T = A \ M , where M is a right ideal of the ring A, then we write N M and gM instead of N T and gT. Let -.~ be the relation on the Cartesian product M • T under which (z, s) ,-. (y, t) .' '.. 3c, d E A

such that x c = y d

and s c = t d E T .

We can immediately verify that .~ is an equivalence relation. Denote by h the natural surjective mapping N • T --, (N • T ) / . ~ - N ( T ) and by v the mapping of N into N ( T ) such that v(n) = h(n, 1). On N ( T ) let us define the operations of addition and of multiplication by elements of Q as follows:

h((x, s)) + h((y, t)) = h((xc + ya, u)), h((x, c)) . f ( b ) f ( t ) -1 : h((xc, tu)),.

where

u=sc=tdGT,

where

b G A,

t E T,

sc = bu,

u G T.

It is immediate that N ( T ) is a right Q-module, v is an A-module h o m o m o r p h i s m , and the mapping w: N T ~ N ( T ) such that w ( n | z) = h((n, 1))z is a Q-module homomorphism, and we have v = g T w . Therefore, we shall identify N T and N ( T ) . 198

2.19. L e m m a . Let Q be the right ring of quotients of a ring A with respect to a set of right denominators T, let N A be a module over A, and let f: A ~ Q and g: N ~ N T be the canonical homomorphisms. Then the following assertions hold: (1 / Ker(g) = { n E N t : I t e T : n t = 0 } and N/Ker(g / T; (2) ( F Jr G)T = FT Jr GT and (F N G)T = FT N GT for all F , G E Lat(NA);

(3) H = for all H e Lat((lVT'))Q; (4) if m is a cardinal and H is an m-generated right ideal of the ring Q, then there exists an mgenerated right ideal B of the ring A such that H = B7" ; (5) AQ is a flat module and (F/G)7" ~- F T / G T for all F e Lat(N) and G 9 Lat(F); (6 / if n e N , F e Lat(N), and g(n) e FT, then nt e F for some t e T; (7) if (NT)Q is a distributive module, then (F N (G Jr H))7" = (F N G Jr F M H ) T for all F , G, H e Lat(NA 1 ; (8) if NA is a distributive module, then (NT) Q is a distributive module; (9) if NA is a distributive module and T = A \ M , where M is a right idea/of the ring A , then (NT)Q is a chain module; (10) if NA is a flat (projective) module, then (NT)Q is a fiat (projective) module; (11) if m is a cardinal and all m-generated right ideals of the ring A are flat A-modules, then all m-generated right ideals of the ring Q are flat (projective) Q-modules.

P r o o f . Assertions (1), (2), (3), and (4) can be verified by means of 1.15 and 2.18. Let us prove (5). Let h : L ~ M be a monomorphism of right modules over A, let u : M ~ M r be the canonical module homomorphism, and let x E L such that u(h(z)) = 0. By (1), there exists an element t E T such that h(x)t = 0. Then xt e Ker(h) = 0. Therefore, (xA)T = 0. Then AQ is a flat module. The remaining part of this item follows from the fact that AQ is a flat module and that we have (XT)B ~- ( X | Q)Q for any module X a . Assertion (6) follows from (5) and (1). Assertion (7) follows from (2). Assertion (8) follows from (3 / and (7). Assertion (9) follows from (81, 1.15 (12/, and 1.24 (2). Assertion (10) follows from [10, Proposition 3 from w Assertion (11) follows from (10) and (4). [] 2~ Let A be a right localizable ring all of whose principal right ideals are flat. Then A is a reduced ring, and AM is a local domain for any M E maX(AA). Indeed, let M E max(AA) and Q - AM. By 1.15 (12) and 2.19 (11), Q is a local ring with flat principal right ideals. By 2.14, Q is a domain. By 1.16 (4), A is a reduced ring. 2.21. L e m m a . Let all 2-generated right ideals of a ring A are flat. Then the following assertions hold: (1) for all elements u , v , w , z E A such that uv = z w , there are elements f , g , h E A such that u / = z g a n d (1 - f)v = hw; (2) for a/l elements u, v, w E A such that uv = vw , there are elements f , g, h E A such that u f = vg and (1 - / ) v = hw; (3) if the ring A is local and B and C are principal right ideals of the ring A and B M C ~ O, then A is a domain and we have B C C or C C B; (4) if A is a right uniform local ring, then A is a right chain domain; (5) if/or any M E max(An), the ring AM exists and is right uniform, then A is a right distributive reduced ring. P r o o f . Assertion (1) follows from 2.17. Assertion (2) follows from (1) for z = w. (3). By 2.14, A i s a d o m a i n . Let B = u A and C = z A . Assume that B ~ C . We must prove the inclusion C C B . By assumption, there are u, v E A such that uv = zw ~ O. By (11, there are elements f , g, h E A such that u f = zg and (1 - f ) v = hw. The element f is not invertible because otherwise we must have u E C and B C C. Since A is local, it follows that 1 - f E U ( A ) . Then v = (1 - f ) - l h w and zw = uv = u(1 - f l - l h w . By the cancellation of the nonzero element w of the domain A, we obtain the relations z = u(1 - f l - l h and C = z A C_ u A = B . We arrive at a contradiction. (41, (5). Assertion (4) follows from (3). Let us prove (5). By (4), 1.15 (12), and 2.19 (11), the ring AM is a right chain domain for any M E max(An). By 1.17, A is a right distributive reduced ring. [] 199

2.22. L e m m a . Assume that A is a right invariant ring and that either all elements o f A with zero square are centra/, or for any a 9 A, there is a positive integer n = n(a) such that r(a n) = r(a'~+a), or A is a ring with the ascending chain condition for the right annihilators, or A is semiprime. Then the following assertions hold: (1) if all principal right ideals of the ring A are fiat, then A is a right localizable reduced ring, and for any M 9 max(An), the ring AM is a right uniform local doma/n; (2) if all 2-generated right ideals of the ring A are fat, then A is a right distributive reduced ring.

P r o o f . (1). It follows from the assumptions that A is a right localizable ring [3]. By (1), A is a reduced ring. Let M 9 max(AA), Q -- AM, and H is the kernel of the canonical homomorphism f : A --, Q. By (1), Q is a local domain. Then H is a completely prime ideal. Therefore, the right invariant domain A / H is right uniform. By 1.15 (10), Q is right uniform.

(2). This follows from (1) and 2.21 (5).

[]

2.23. Let MA be a module over a distributive invariant ring A. All submodules of the module M are flat iff all cyclic submodules of the module M are flat. (This follows from 2.16.) 2.24. P r o p o s i t i o n . For an invariant semiprime ring A, the following conditions are equivalent: (1) all submodules of f a t modules over the ring A are fiat; (2) MI 2-generated right ideals of the ring A are fiat; (3) A is a distributive ring. By 2.23, the implication (1) ==~ (2) is obvious, the implication (2) the implication (3) '.. (1) follows from 2.23, 2.5 (2) and 2.7.

'.. (3) follows from 2.22 (2), and

2.25. Let all endomorphisms of principal right ideals of the ring A be extendable to endomorphisms of the module AA. Then the following assertions hold: (I) if a 9 A and if r(a) is an ideal of the ring A (for example, this is the case if a is a right regular element of the ring A), then Aa is an ideal of the ring A; (2) if in the ring A, the right annihilators of all elements are ideals (for example, this is the case if A is a reduced or a right invariant ring), then the ring A is left invariant. It suffices to prove (1) because (2) follows from (1). Let b 9 A. Since r(a) is an ideal, we have r(a) C_ r(ab), and there exists an epimorphism f : aA ---* abA such that f(a) = ab. Therefore, f 9 End(aA). By assumption, there exists an element t 9 A such that ta = f(a) = ab. Hence, Aa is an ideal of the ring A. 2.26. Let any idempotent endomorphism of any 2-generated submodule of the module MA be extendable to an endomorphism of the module M , let F be a completely invariant submodule of the module M , and let G1 and G~ be submodules of the module M such that G1 N G2 = 0. Then F N (G1 + G2) = F N GI + F n G2, and for all elements gl 9 G1 and g2 9 G~, there are endomorphisms dl and d2 of the module M such that dl(Y,) = d2(g2) = O,

dl(g2) = g2,

d2(gl) = gl.

Let H - glA 9 g2A and let hi: H --* giA be the natural projections. The endomorphisms hi can be extended to endomorphisms d~ E End(M). Then d,(gl)=d2(g2)=O,

dl(g2)=g2,

d2(gl)=gl.

If f = gl + g2 E F n ( G 1 + G2), then g, = d2(gl + g2) = d2(f) and g~ = d l ( f ) . Hence, di(f) E F because F is a completely invariant submodule of the module M . Therefore, f=d2(f)+d~(f)EFNG~

200

+FNG2

and

FN(GI+G2)

C_FNG~ + F N G 2 C _ F N ( G ~ + G 2 ) .

2.27. Let any i d e m p o t e n t endomorphism of any 2-generated right ideals of a ring A can be extended to an e n d o m o r p h i s m of the m o d u l e AA, let F be an ideal of the ring A, and let G1 a n d G2 be right ideals of the ring A such t h a t G1NG2 = 0. T h e n the following assertions hold: (1) F N ( G 1 + G 2 ) = F N G 1 + F N G 2 , and for any elements gl E G1 and g2 E G2, there are elements dl, d2 E A such t h a t dig1 = d2g2 = 0, dig2 = g2, and d2(gl) = gl ; (2) if an element a E A is right regular, then a A is an essential right ideal. Assertion (1) follows from 2.26. (2). Let b 9 A , a A n bA = 0. By (1), there exists an d e m e n t d E A such that da = 0 and db = b. T h e n d E g(a) = 0 and b = db = O. 2.28. Let any e n d o m o r p h i s m of any 2-generated right ideal of an arbitrary quotient ring B of the ring A be extendable to an e n d o m o r p h i s m of the module BB. T h e n the following assertions hold: (1) A is an arithmetical ring; (2) if in a ring A, the right annihilators of all elements are ideals (for example, this is the case if A is a reduced ring), then A is a left invariant left distributive ring; (3) if A is right invariant, then A is an invariant distributive ring. Let F , G, a n d H be ideals of the ring A such that F C_ G + H , let M - F f 3 G + F f3 H , and let h: A --* A / ( G f3 H ) be the natural epimorphism. By 2.27 (1), applied to h ( A ) we have h(r) = h(f)N

h(G) + h ( F ) N h ( H ) .

Then, by applying the m o d u l e law, we see that F+GNH=M+GNH

and

F=Ff3(M+G)NH=M+FNGf3H=M,

and assertion (1) is thus proved. Assertion (2) follows from (1) and 2.25 (2). Assertion (3) follows from (2). 2.30. Let Ax a n d A2 be ideals of the ring A and let xl and z2 be elements of the ring A such that xl 3:2 E A1 + A2. T h e n there exists an element a E A such that a - 3:1 9 A1 and a - z2 9 A2. (Let 3:1 - - X 2 = a l - - a2, where al E A1, a2 E A2, and a - 3:1 - al = 3:2 - a 2 ; t h e n a - 3:1 = - a l E Ax and a - 3:2 = - a 2 E A2, and thus a is the desired element.) 2.31. L e m m a . T h e following conditions are equivalent: (1) A is an arithmetical ring; (2) for a n y i d e a l s AI , . . . , A," of the ring A a n d any elements xl , . . . , 3:," E A such that 3 : i - x I E A i + A 1 [oral1 i , j , r an element 3: E A such that 3 : - 3:i E Ai for i = 1, . . . , n ; (3) for any ideals A 1 , A2, and A3 o f the ring A and for any element d 9 (A1 + A2) M ( A 1 + Aa), there ex/sts an element 3: E A1 such that 3: - d 9 A2 N A a . P r o o f . T h e implication (2) ==~ (3) follows from the fact that (2) becomes (3) for n = 3, 3:1 = 0, and 3:2 =3:3 = d .

(1) '.. (2). For n = 2, the assertion follows from 2.30. Suppose the assertion holds for n - 1. There is an element b 9 A such t h a t b - 3:i 9 Ai for i = 1, . . . , n - 1. Moreover, we have xl - 3:,, 9 Ai + A,, for i = 1, ... , n - 1, and thus

N (A, + A,').

b-3:,'=(b-3:i)+(x,-3:,') 9

i=1 ,'--1

Since A is an a r i t h m e t i c a l ring, it follows t h a t b - z," 9 A , + Ni=l Ai a n d b - 3:, = a , - d, where a," 9 A , and d 9 N~-~ Ai. Let a = b + d. T h e n a - 3:," = an 9 A,'. Moreover, a - xi = b - xi -t- d 9 Ai for i = 1, . . . , n - 1. Therefore, a is the desired element. (3) '.. (1). It suffices to prove that if M , B , and C are ideals of the ring A a n d if dE(M+B)

M(M+C),

then

dEM+(BMC).

Let d = m l + b = m2 + c, where m l , m2 9 M , b E B , and c E C . By assumption, there exists an element 3: E M such t h a t 3: - d E B fq C . Let y - d - 3: E B fl C . T h e n d = 3: + y E M + B M C . [7 201

2.32. Completely closed modules and rings. A module M is said to be completely closed if all endomorphisms of its ~nltely generated submodules can be extended to endomorphisms of the module M . A ring A is completely closed if and only if for any right ideal B of the ring A and any endomorphism f E End(BA), there exists an element a E A such that f(b) = ab for all b E B . A commutative domain A is completely closed, as a module over itself, in the sense of the above definition, if and only if A is completely closed in the classical sense in its own field of quotients. 2.33. L e m m a . For a ring A, the following conditions are equivalent: (1) A is a distributive invariant ring; (2) A is an invariant ring all of whose quotient rings are completely closed rings; (3) all quotient rings of the ring A are completely dosed rings, and in the ring A, the right annihilators and left annihilators of all eIements are ideals. P r o o f . The equivalence of conditions (2) and (3) and the implication (2) ~ (1) follow from 2.28. Let us prove the implication (1) - - ~ (2). Since condition (1) is symmetric and is inherited by quotient rings, it suffices to prove that the module AA is completely closed. Let n

~Ami

=MCA.

i=l

Since A M is a finitely generated distributive module over an invariant ring A, it follows from 2.29 that f ( A m i ) C Arni for all i. Therefore, there exist elements x l , . . . , x,~ E A such that f(rni) = rnixi E m i A = Arni. Then 0 = (Am, M Amj)(xi - xj) Vi,j. Since Arni N m i A f3 rnjA, we have xl - xj =_ dij E r(miA N m j A ) . By 4.2, there are elements aij, bij E A such that I = alj + bij and miAaij + mjAbl j C_m i A N m j A . Then aijdij E r(miA) - Ai, bir fi r(mjA) = A j , and zi - zj = aijdij + biidij fi Ai A- A i 9 By 2.31, there exists an dement z q A such that z - zi E Ai for all i. Then an endomorphism g E End(AA) that is an extension of f can be (weft) defined by the rule g(y) = yz for y E A. [] 2.34. P r o p o s i t i o n . For a reduced ring A, the following conditions are equivalent: (1) all quotient rings of the ring A are completely dosed rings; (2) A is an invariant ring over which all submodules off/at modules are fiat. Since in a reduced ring, by 1.11, all right annihilators and all left annihilators are ideals, assertion 2.34 follows from the fact that in the case of a reduced ring, both (1) and (2) are equivalent to the fact that A is an invariant distributive ring (see 2.33 and 2.24). [] 2.35. P r o p o s i t i o n . For a ring A, the following conditions are equivalent: (1) A is a right distributive semiprime ring and any quotient ring of A by a minimal prime ideal is a right Noetherian left finite-dimensional ring; (2) A is a left distributive semiprime ring and any quotient ring of A by a minimal prime ideal is a left Noetherian right finite-dimensional ring; (3) A is an invariant reduced ring over which all submodules of fiat modules are fiat, and all prime quotient rings of the ring A are invariant hereditary Noetherian domains. P r o o f . Since condition (3) is symmetric, it suffices to prove the equivalence of conditions (1) and (3). The implication (3) ==~ (1) follows from Proposition 2.24. Let us prove the implication (1) ==* (3). By Lemma 1.17 (6), A is an invariant reduced ring whose any prime quotient ring B is an invariant distributive Noetherian domain that has skew field of quotients. Now we apply assertion 2.24 to the ring A and assertions 2.24 and 2.12 to the ring B. [] 202

References 1. C. U. Jensen, "A remark on arithmetical rings," Proc. Amer. Math. Sot., 15, No. 6, 951-954 (1964).

2. 3. 4. 5. 6. 7. 8. 9. 1O. 11.

C. Faith, Algebra: Rings, Modules, and Categories. Corrected reprint, Vol. 1, Springer-Verlag, Berlin-New York (1973). A. A. Tuganbaev, "Left and right distributive rings," Mat. Zametki [Math. Notes], 58, No. 4, 604-627 (1995). L. H. Rowen, Ring Theory, Student edition, Academic Press, Boston (1991). H. Bass, Lectures on Topics in Algebraic K-Theory, "rata Inst. of Fundamental Research, Bombay (1967). N. Bourbaki, Commutative Algebra. Chapters 1-7 [English translation], Springer-Verlag, Berlin-New York (1989). F. Kasch, Modulen und Ringe, Teubner, Stuttgart (1977). R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia (1991). B. Stenstri~m, Rings of Quotients: an Introduction to Methods of Ring Theory, Springer Verlag, Berlin (1975). J. Lambek, Lectures on Rings and Modules, Blaisdell, Waltham, Mass.-Toronto, Ont.-London (1966) N. I. Dubrovin, The Rational Closure of Group Rings of Left-Ordered Groups, Gerhard Mercator Universit~t Duisburg Gesamthochschule (1994).

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