To each associative ring R we can assign the adjoint Lie ring R(-) (with the operation (a ... It is clear that a Lie ring R(-) is commutative if and only if the semigroup.
Mathematical Notes, Vol. 6~, No. 4, 1997
On the Semigroup Nilpotency and the Lie Nilpotency of Associative Algebras A. N. K r a s i l t n i k o v
UDC 512.552, 512.532
ABSTRACT. To each associative ring R we can assign the adjoint Lie ring R ( - ) (with the operation (a, b) = ab - ba) and two semigroups, the multiplicative semigroup M(R) and the associated semigroup A(R) (with the operation a o b ----ab -b a + b). It is clear that a Lie ring R ( - ) is commutative if and only if the semigroup M(R) (or A(R)) is commutative. In the present paper we try to generalize this observation to the case in which R ( - ) is a nilpotent Lie ring. It is proved that if R is an associative algebra with identity element over an infinite field F , then the algebra R ( - ) is nilpotent of length c if and only if the semigroup M ( R ) (or A(R)) is nilpotent of length c (in the sense of A. I. Mal~tsev or B. Neumann and T. Taylor). For the case in which R is an algebra without identity element over F , this assertion remains valid for A ( R ) , but fails for M ( R ) . Another similar results are obtained. KEY WORDS: nilpotent Lie ring, nilpotent semigroup, associated semigroup.
To each associative ring R we can assign the adjoint Lie ring R (-) (with the operation (a, b) --- ab-ba) and two semigroups, the multiplicative semigroup M(R) and the associated semigroup A(/~) (with the operation a o b --- ab + a -k b). If R is a ring with identity element, then the semigroups M( R) and A( R) are isomorphic (the mapping ~: r --* r + 1 is an isomorphism of A(R) onto M(R)); however, if R is a ring without identity element, then M(R) and A(R) are not isomorphic (in this case the semigroup M(R) does not have an identity element, while the semigroup A(R) does). It is clear that the Lie ring R(-) is commutative if and only if the semigroup M(R) (or A(R)) is commutative. Can this fact be generalized to the case in which R (-) is a Lie ring? For a generalization of the commutativity condition to semigroups, we can take the notions of nilpotency in the sense of Mal'tsev [1] or Neum~nn and Taylor [2] or the notion of strict nilpotency introduced by Shpil'rain [3] (the exact definitions of these notions are given below). In general, the notions listed above are pairwise nonequivalent, but they turn out to be equivalent for the semigroups treated in the present paper. In what follows, unless otherwise stated, we understand the nilpotency of a semigroup in the sense of any of the papers [1-3]. For an associative ring R, the following problems arise. P r o b l e m 1. Let M ( R ) (or A(R)) be a nilpotent semigroup of length c. Is the Lie ring R(-) nilpotent (and if so, then what is its nilpotency length)?
P r o b l e m 2. Let R (-) be a nilpotent Lie ring of length c. Is the semigroup M(R) (or A(//)) nilpotent (and if so, then what is its nilpotency length)? An exhaustive answer to both questions can be obtained for the case in which R is an algebra over an infinite field. If R is an algebra with identity element, then the answer for the semigroup M(R) is as follows. T h e o r e m 1. Let R be an a/gebra with identity element over an in6nite ~eld. Then the following assertions hold. 1) /_f M ( R ) is a nilpo~ent sere/group of length c, then the Lie a/gebra R (-) is nilpotent of leng2h c. 2) /_f R(-) is a ns Lie Mgebra of leng2h c, then the semigroup M(R) is nilpotent oflength c. 9 ranslated from Matematicheskie Zametki, Vol. 62, No. 4, pp. 510-519, October, 1997. Original article submitted March 26, 1996. 426
0001-4346/97/6234-0426 $18.00
~)1998 Plenum Publishing Corporation
The assertion of T h e o r e m 1 fails for algebras without identity element. Namely, in general, the answer to Problem 1 for the semigroup M ( R ) associated to such algebras is negative. This is shown by the following assertion. 2. Let F be a tleld of characteristic 2, let F[ti [ i E ~ be the polynomial algebra over F , and let I be the ideal of F[ti [ i E ~ generated by dements of the form t~ ( i E N). Let R = F[ti [ i E N~/I, let M be a free cyclic module over R, and let Proposition
o) be a matrix ring. In this case the semigroup M(R) is nilpotent of length 2, but the Lie a/gebra R ( - ) is not nilpotent. For an associative algebra R over an infinite field (with or without an identity element), an exhaustive answer to Problems 1 and 2 for the semigroup A(R) is as follows. Theorem
3. Let R be an a/gebra over an irr~nite field (not necessarily with an identity elemen@
Then the following assertions hold. 1) / f the sern/group A( R) is nilpotent of length c, then the Lie a/gebra R (-) is nilpotent oflength c. 2) / f the Lie Mgebra R(-) is nilpotent of length c, then the semigroup A(R) is nilpotent oflength c. This t h e o r e m is obtained as a consequence of Propositions 4 and 5 (see below) a n d generalizes Theorem 1 because, for an algebra R with identity element, the semigroups A(R) and M ( R ) are isomorphic. For arbitrary associative rings, an exhaustive answer to Problem 2 is possible (both for the semigroup M(R) and for A ( R ) ) . As far as Problem 1 is concerned, the answer is obtained in some special cases only. The answer to P r o b l e m 2 is as follows. P r o p o s i t i o n 4. Let R be an associative ring such that R(-) is a nilpotent Lie ring of length c. /n this case the semigroups M(R) and A(R) are nilpotent of length at most c. N. G u p t a a n d F. Levin [4] proved that if the associated Lie ring R (-) of an associative ring R is nllpotent of length c, t h e n the group of units, U(R), of the ring R is also nilpotent of length at most c (somewhat later, an equivalent assertion was independently proved by H. Laue [5]). Proposition 4 generalizes this result because a group G is a nilpotent semigroup of length c if and only if G is a nilpotent group of length c. However, we note that the Gupta-Levin theorem is heavily used in the proof of Proposition 4. Proposition 4 and Theorem 3 were first nn - o u n c e d in [6]. In some cases, an answer to Problem 1 for the semigroup A(R) is given by t h e following proposition (for the dei~nltions and the main properties of the varieties of associative rings a n d free rings of varieties,
see [7, 8]).
Proposition 5. Let R be an associative ring such that the Tree rings of the variety generated by R are residually nilpotent, i.e., if B is a Tree ring (without identity element) of the variety generated by R, then N~=l Bn = {0}. Let the semigroup A(R) be nilpotent of length c. /n tMs case the Lie ring R(-) is nilpotent of length c. T h e o r e m 3 is an obvious consequence of Propositions 4 and 5. These propositions imply the following assertion as well. C o r o l l a r y 6. Let R be a nilring or a ring w/thout additive torsion for which the associated semigroup A(R) is nilpotent of length c. /n t/z/s case the Lie ring R (-) is nilpotent of length c. Let G be a locally finite p-group, and let F be a field of prime characteristic p. Since the fundamental ideal of the group algebra F G is locally nilpotent, it follows that Corollary 6 implies the next statement. 427
C o r o l l a r y 7 [9]. Let G be a locally finite p-group, and let F be a field ofprime characteristic p. In th/s case the associated Lie algebra FG(-) of the group algebra FG is nilpotent of leng2h c if and on/y if the group of units, U ( F G ) , of the a/gebra FG is nilpo~ent of length c. We present another special case in which the answer to Problem 1 for the semigroup A(R) is positive.
Proposition 8. Let R be an associative ring whose associated semigroup A( R) is nilpotent of length 2. In this case the Lie ring R(-) is nilpotent of length 2. We stress that in the general case (for associative rings that do not satisfy the assumptions of Proposition 5) the following problem remains open. P r o b l e m . Is it true that the associated Lie ring R(-) of an associative ring R is nilpotent of length c if and only if the associated semigroup A(R) is nilpotent of length c in the sense of one of the papers [1-3]? If to the statement of this problem we add the assumption that R is a radical ring in the sense of Jacobson (in this case, A(R) is a group), then we obtain the Jennings problem [10] that was treated in [10, 5, 11] and completely solved in [9]. R e m a r k 1. If R is an algebra over a field of characteristic 0 and if the semlgroup M(R) is nilpotent, then it follows from the paper of Golubchik and Mikhalev [12] that the Lie algebra R(-) has the Engel property of bounded length, and hence is nilpotent [13], and, in general, its nilpotency length exceeds that of M ( R ) . R e m a r k 2. A semigroup is said to be Ma?~ev niIpotent of length c [1] if it satisfies the identity Uc+l = Ve+I but does not satisfy the identity Uc = Vc,
where
U0 = x,
V0 = y,
Uk+l = UkzkVk,
Vk+l = VkzkUk
(k = 1 , 2 , . . . ) ,
and x, y, z0, z l , . . , are variables. If in this definition we set z0 = 1, then we obtain the definition of a Neumann-Taylor nilpotent semigroup of length c [2]. Finally, a semigroup is Shpi~ rain strictly nilpotent of length at most c [3] if it satisfies all semigroup identities of a free nilpotent group of length c of countable rank. As mentioned above, the various notions of nilpotency of a semigroup are pairwise not equivalent [14]. It immediately follows from the results of [1, 2] that a strictly nilpotent sernigroup of length c is nilpotent of length c both in the sense of [1] and [2]. It is clear that a semigroup with identity element that is nilpotent of length c in the sense of [1] is nilpotent of length at most c in the sense of [2]. We note that the semigroups treated in the present paper that are nilpotent of length c in the sense of one of the papers [1-3] are nilpotent of length c in the sense of the other papers as well. Introduce the notation (x, y) = xy -- yx,
[x, y] = x - l y - l x y ,
( z l , . . . ,xk, z k + x ) = ((xa, ... , x/~), zl:+l), [ z l , . . . , Zk,Zk+l]
=
[ [ x l , . . . ,Xk],Zk+l],
k=2,3,
....
P r o o f o f P r o p o s i t i o n 2. By Remark 2, it suffices to prove that the semigroup M(R) satisfies all semigroup identities of a free nilpotent group G of length 2 of countable rank, i.e., the identities of the group G of the form =
x;:
...
where rni, n i >_O. Lemma
1. Let
e
(i = 1 , . . . , k + 1),
I n tI~$ c a s e ~'1 "'" FkFk+l m_ O.
428
let
=
for some i l , i2 ( 1 _< il
