Proc. Indian Acad. Sci. (Math. Sci.) Vol. 117, No. 2, May 2007, pp. 167–175. © Printed in India
Derivations and generating degrees in the ring of arithmetical functions ALEXANDRU ZAHARESCU and MOHAMMAD ZAKI Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA E-mail:
[email protected];
[email protected] MS received 26 May 2005 Abstract. In this paper we study a family of derivations in the ring of arithmetical functions of several variables over an integral domain, and compute the generating degrees of the ring of arithmetical functions over the kernel of these derivations. Keywords.
Derivations; generating degrees; arithmetical functions.
1. Introduction Cashwell and Everett [5] proved that the ring (A, +, ·) of complex-valued arithmetical functions with Dirichlet convolution is a unique factorization domain. Yokom [10] studied the prime factorization of arithmetical functions in a certain subring of the regular convolution ring, and determined a discrete valuation subring of the unitary ring of arithmetical functions. Recently, Alkan et al [4] investigated a class of derivations and norms in the ring of arithmetical functions in several variables with unitary convolution. Schwab and Silberberg [8] constructed an extension of (A, +, ·) which is a discrete valuation ring. In [9], they showed that A is a quasi-noetherian ring. Further results, in a more abstract categorical setting, have been obtained by Schwab in [6] and [7]. In [3], a class of absolute values and a family of derivations on the ring of arithmetical functions in several variables, with the analogue of Dirichlet convolution as multiplication, are defined and studied. Let R be an integral domain, r a positive integer, and Ar (R) = {f : Nr → R}. For any f, g ∈ Ar (R), the convolution f ∗ g of f and g is defined by n1 nr (f ∗ g)(n1 , . . . , nr ) = . ··· f (d1 , . . . , dr )g ,..., d1 dr d |n d |n 1
1
r
r
The topologies obtained from the valuation constructed in [8] and its generalizations defined in [3] play an important role in our present investigation. We consider a natural family of subrings Br,k,p (R) of Ar (R), which are closed in each of these topologies. For any k ∈ {1, . . . , r}, and any prime number p, Br,k,p (R) consists of all the functions f ∈ Ar (R) with the property that for all n1 , . . . , nr ∈ N with p dividing nk , one has f (n1 , . . . , nr ) = 0. In order to understand the algebraic and topological structure of the extensions Ar (R)/Br,k,p (R), we proceed to find the generating degrees of Ar (R) with respect to each of the subrings Br,k,p (R). For two commutative topological rings A ⊆ B, a subset M ⊆ B is said to be a (topologically) generating set of B over A provided that 167
168
Alexandru Zaharescu and Mohammad Zaki
the ring A[M] is dense in B. Suppose B has a finite generating set M over A. Then the smallest nonnegative integer m for which there exists a generating set M of B over A having exactly m elements, is called the generating degree of B over A, and is denoted by gdeg(B/A) [2]. Thus A is dense in B if and only if gdeg(B/A) = 0. As an example of a nonzero, finite generating degree, if Cp denotes the topological closure of the algebraic closure of the field Qp of p-adic numbers, then gdeg(Cp /Qp ) = 1 [1, 2]. If B has no finite generating set over A, one writes gdeg(B/A) = ∞. For some general connections between generating degrees, continuous derivations, and topological Krull dimensions, the reader is referred to [2]. In the present paper we are also lead to consider a family of continuous derivations θk,p : Ar (R) → Ar (R), and obtain the subrings Br,k,p (R) as kernels of these derivations. We also make use of a topological isomorphism that generalizes the one constructed in [5]. We find that the generating degree of Ar (R) over each of the subrings Br,k,p (R) is equal to 1.
2. Absolute values Let r be a positive integer, R be an integral domain with identity 1R , and Ar (R) = {f : Nr → R}. Note that R has a natural embedding in the ring Ar (R), and Ar (R) with addition and convolution defined as in the Introduction naturally becomes an R-algebra. We now recall the construction from [3] of a class of absolute values on Ar (R). Fix t = (t1 , . . . , tr ) ∈ Rr with t1 , . . . , tr linearly independent over Q, and ti > 0, (i = 1, 2, . . . , r). Given n ∈ N, denote by (n) the total number of prime factors of n, counting multiplicities. Thus, if n = p1α1 . . . pkαk is the prime factorization of n, then (n) = α1 + · · · + αk . Define also r : Nr → Nr by r (n1 , . . . , nr ) = ((n1 ), . . . , (nr )). For any f ∈ Ar (R) denote supp(f ) = {n ∈ Nr |f (n) = 0}, and let Vt (f ) =
min
n∈supp(f )
t · r (n),
with the convention min(∅) = ∞, where t · r (n) = t1 (n1 ) + · · · + tr (nr ). It is shown in [3] that for any f, g ∈ Ar (R) one has Vt (f + g) ≥ min({Vt (f ), Vt (g)}) and Vt (f ∗ g) = Vt (f ) + Vt (g). Next, one extends the valuation Vt to a valuation V¯t on the field of fractions K = { fg |f, g ∈ Ar (R), g = 0} of Ar (R) by letting Vt ( fg ) = Vt (f ) − Vt (g). Further, to obtain an absolute value on K, fix a real number ρ ∈ (0, 1) and define | · | = | · |t : K → R by ¯
|x|t = ρ Vt (x) if x = 0,
and
|x|t = 0 if x = 0.
The above absolute value is nonarchimedean. In [3] it is proved that for any integral domain R, any positive integer r, and any t as above, Ar (R) is complete with respect to the absolute value | · |t .
Ring of arithmetical functions
169
3. Derivations For each integer i with 1 ≤ i ≤ r and each prime number p we define a derivation θi,p = θr,i,p (R) on Ar (R) as follows. For n ∈ N, let vp (n) denote the largest integer m for which p m divides n. Given an f ∈ Ar (R), define θi,p f ∈ Ar (R) by θi,p f (n1 , . . . , nr ) = (1 + vp (ni ))f (n1 , . . . , ni−1 , pni , ni+1 , . . . , nr ), for all (n1 , . . . , nr ) ∈ Nr . Lemma 1. For any f, g ∈ Ar (R), θi,p (f + g) = θi,p f + θi,p g,
(3.1)
θi,p (f ∗ g) = θi,p f ∗ g + f ∗ θi,p g.
(3.2)
and
Proof. The equality (3.1) is clear from the definitions. In order to prove (3.2), fix f, g ∈ Ar (R) and let (n1 , . . . , nr ) ∈ Nr . Write ni as ni = pk m, with m, p relatively prime. Then, θi,p (f ∗ g)(n1 , . . . , nr ) = (1 + vp (ni ))(f ∗ g)(n1 , . . . , ni−1 , pni , ni+1 , . . . , nr ) = (k + 1) ··· ··· d1 |n1
di−1 |ni−1 di |pk+1 m di+1 |ni+1
dr |nr
n1 ni−1 pk+1 m nr f (d1 , . . . , di−1 , di , di+1 , . . . , dr )g ,..., , ,..., d1 di−1 di dr = ··· ··· d1 |n1
di−1 |ni−1 di+1 |ni+1
dr |nr di |pk+1 m
(k + 1)f (d1 , . . . , di−1 , di , di+1 , . . . , dr )g
n1 ni−1 pk+1 m nr ,..., , ,..., d1 di−1 di dr
.
The innermost sum above equals (k+1)
f (d1 , . . . , di−1 , di , di+1 , . . . , dr )g
di |m
+ (k+1)
di |m
n1 ni−1 pk+1 m nr ,..., , ,..., d1 di−1 di dr
f (d1 , . . . , di−1 , pdi , di+1 , . . . , dr )g
n1 ni−1 pk m nr ,..., , ,..., d1 di−1 di dr
+ ··· + (k+1)
di |m
f (d1 , . . . , di−1 , pk+1 di , di+1 , . . . , dr )g
n1 ni−1 m nr ,..., , ,..., d1 di−1 di dr
.
170
Alexandru Zaharescu and Mohammad Zaki
Also, (θi,p f ∗ g)(n1 , . . . , nr ) = ··· d1 |n1
···
di−1 |ni−1 di |pk m di+1 |ni+1
dr |nr
θi,p f (d1 , . . . , di−1 , di , di+1 , . . . , dr )g =
d1 |n1
···
···
di−1 |ni−1 di+1 |ni+1
n1 ni−1 pk m nr ,..., , ,..., d1 di−1 di dr
dr |nr di |pk m
(1+vp (di ))f (d1 , . . . , di−1 , pdi , di+1 , . . . , dr )g Here the innermost sum equals
f (d1 , . . . , di−1 , pdi , di+1 , . . . , dr )g
di |m
+
n1 ni−1 pk m nr ,..., , ,..., d1 di−1 di dr
n1 ni−1 pk m nr ,..., , ,..., d1 di−1 di dr
2
2f (d1 , . . . , di−1 , p di , di+1 , . . . , dr )g
di |m
.
n1 ni−1 pk−1 m nr ,..., , ,..., d1 di−1 di dr
+ ... n1 ni−1 m nr + . (k+1)f (d1 , . . . , di−1 , pk+1 di , di+1 , . . . , dr )g ,..., , ,..., d1 di−1 di dr d |m i
Similarly, (f ∗ θi,p g)(n1 , . . . , nr ) = ··· d1 |n1
di−1 |ni−1 di+1 |ni+1
···
dr |nr di |pk m
k p m g f (d1 , . . . , di−1 , pdi , di+1 , . . . , dr ) 1 + vp di n1 ni−1 pk+1 m nr , × ,..., , ,..., d1 di−1 di dr and the innermost sum equals
(k + 1)f (d1 , . . . , di−1 , di , di+1 , . . . , dr )g
di |m
+
di |m
n1 ni−1 pk+1 m nr ,..., , ,..., d1 di−1 di dr
kf (d1 , . . . , di−1 , pdi , di+1 , . . . , dr )g
n1 ni−1 pk m nr ,..., , ,..., d1 di−1 di dr
+ ... n1 ni−1 pm nr + . f (d1 , . . . , di−1 , pk di , di+1 , . . . , dr )g ,..., , ,..., d1 di−1 di dr d |m i
Ring of arithmetical functions
171
By comparing the innermost sums in the above expressions one finds that θi,p (f ∗ g) = θi,p f ∗ g + f ∗ θi,p g. Theorem 1. Let R be an integral domain and let r be a positive integer. For any prime number p and any k ∈ {1, . . . , r}, θk,p is a derivation on Ar (R) which is continuous with respect to each absolute value of the form | · |t on Ar (R). Proof. By Lemma 1 we know that θk,p is a derivation on Ar (R). In order to show that θk,p is continuous with respect to each absolute value of the form | · |t on Ar (R), note first that for any f ∈ Ar (R), |θk,p f |t ρ Vt (θk,p f ) = |f |t ρ Vt (f ) = ρ Vt (θk,p f )−Vt (f ) . Let n1 , . . . , nr be such that Vt (θk,p f ) = t1 (n1 ) + · · · + tr (nr ). So in particular (n1 , . . . , nr ) ∈ supp(θk,p f ). By the definition of θk,p f it follows that (n1 , . . . , nk−1 , pnk , nk+1 , . . . , nr ) ∈ supp(f ). Thus, Vt (f ) ≤ t1 (n1 ) + · · · + tk−1 (nk−1 ) + tk (nk ) + tk+1 (nk+1 ) + · · · + tr (nr ) = Vt (θk,p f ) + tk . Therefore tk ≥ Vt (f ) − Vt (θk,p f ), which in turn implies that ρ Vt (θk,p f )−Vt (f ) ≤ deduce that for all f ∈ Ar (R), |θk,p f |t ≤
1 ρ tk
. We
1 |f |t . ρ tk
This shows that θk,p is a bounded linear operator on Ar (R), and hence it is continuous.
4. The ring of formal r -fold power series In this section we discuss a generalization to r variables of the isomorphism constructed by Cashwell and Everett in [5]. Let R be an integral domain. Let p1 , p2 , p3 , . . . denote the sequence of prime numbers in increasing order. Then every integer n may be writα (n) α (n) ten uniquely in the form n = p1 1 p2 2 . . . and uniquely described by a vector (α1 (n), α2 (n), . . . ) with non-negative integral components, finitely many of which are nonzero, all such vectors being realized as n ranges over N. Define a 1-fold formal power series in a countable number of indeterminates yp1 , yp2 , yp3 , . . . with coefficients in R, by cn ypα11 (n) ypα22 (n) . . . . n α (n) α (n) α (n)
Here the summation extends over all n = p1 1 p2 2 p3 3 . . . in N, and cn belongs to R for any such n. More generally, a formal r-fold power series in r distinct countable sets
172
Alexandru Zaharescu and Mohammad Zaki
of indeterminates x1p1 , x1p2 , . . . , x2p1 , x2p2 , . . . , xrp1 , xrp2 , . . . with coefficients in R has the form α (n ) α (n ) α1 (nr ) α2 (n1 ) α2 (n2 ) α2 (nr ) ··· cn1 ,...,nr x1p1 1 1 x2p1 1 2 . . . xrp x1p2 x2p2 . . . xrp ..., 1 2 n1
nr
where the summation extends over all α (n1 ) α2 (n1 ) α3 (n1 ) p2 p3 ...,
n1 = p1 1
α (n2 ) α2 (n2 ) α3 (n2 ) p2 p3 ...,
n2 = p1 1
... α (nr ) α2 (nr ) α3 (nr ) p3 ... p2
nr = p1 1
in N, and cn1 ,...,nr ∈ R. For a fixed positive integer r, we denote the ring of all such r-fold power series by R{. . . , xipj , . . . } = R{x1p1 , x1p2 , . . . }{x2p1 , x2p2 , . . . } . . . {xrp1 , xrp2 , . . . }. The multiplicative operation in this ring is the usual formal operation on power series involving multiplication and collection of finite number of like terms. We emphasize that only a finite number of xipj actually appear (in the sense that αj (ni ) > 0) in any term. However, infinitely many xipj may occur in the same series. For any positive integer l, any k ∈ {1, . . . , r}, and any power series Q ∈ R{. . . , xipj , . . . }, denote by degxkp (Q) the supremum of the set of exponents of xkpl that appear in the terms of Q l with nonzero coefficients. Also, for a positive integer l, and k ∈ {1, . . . , r}, denote by R{. . . , xipj , . . . }(i,pj )=(k,pl ) the subring of R{. . . , xipj , . . . } which consists of all power series Q in R{. . . , xipj , . . . } such that degxkp (Q) is zero. Consider now the followl ing class of absolute values on R{. . . , xipj , . . . }. Fix a real number 0 < ρ < 1 and a t = (t1 , . . . , tr ) with t1 , . . . , tr > 0, and t1 , . . . , tr linearly independent over Q. For any Q ∈ R{. . . , xipj , . . . }, Q=
···
n1
nr
α (n1 ) α1 (n2 ) α1 (nr ) α2 (n1 ) α2 (n2 ) α2 (nr ) x2p1 . . . xrp x1p2 x2p2 . . . xrp ..., 1 2
cn1 ,...,nr x1p1 1
denote supp(Q) = {n ∈ Nr |cn1 ,...,nr = 0}, and let Wt (Q) =
min
n∈supp(Q)
t · r (n).
Then one obtains an absolute value | · | t on R{. . . , xipj , . . . } by letting |x| t = ρ Wt (x) if x = 0,
and
|x| t = 0 if x = 0.
Lemma 2. For any prime number pj0 and any k ∈ {1, . . . , r}, xkpj0 is a generating element of R{. . . , xipj , . . . } over R{. . . , xipj , . . . }(i,pj )=(k,pj0 ) with respect to each absolute value || t . The proof follows easily by writing each element Q of R{. . . , xipj , . . . } as a formal power series in the variable xkpj0 with coefficients in R{. . . , xipj , . . . }(i,pj )=(k,pj0 ) and then m , for each positive integer m. In this way one obtains truncating this power series up to xkp j0 for each m a polynomial Qm in xkpj0 with coefficients in R{. . . , xipj , . . . }(i,pj )=(k,pj0 ) . As
Ring of arithmetical functions
173
m tends to infinity, Qm converges to Q in the absolute value |x| t . This follows from the fact that the tail Q−Qm consists only of terms in which xkpj0 appears with exponents larger than m, and hence Wt (Q − Qm ) ≥ mtk , which tends to infinity as m → ∞. In conclusion the ring of polynomials in the variable xkpj0 with coefficients in R{. . . , xipj , . . . }(i,pj )=(k,pj0 ) is dense in R{. . . , xipj , . . . }, which proves the lemma. Next, following [5], we associate to each arithmetical function a = a(n) ∈ A1 (R) in one variable a formal power series in a countable number of indeterminates yp1 , yp2 , yp3 , . . . with coefficients in R, by means of the correspondence a(n)ypα11 (n) ypα22 (n) . . . . a → P (a) = n α (n) α (n) α (n)
Here the summation extends over all n = p1 1 p2 2 p3 3 . . . in N. Similarly, for any r ≥ 1, an arithmetical function a = a(n1 , . . . , nr ) ∈ Ar (R) in r variables may be associated to a formal r-fold power series in a countable number of indeterminates x1p1 , x1p2 , . . . , x2p1 , x2p2 , . . . , . . . , xrp1 , xrp2 , . . . with coefficients in R, by means of the correspondence α (n ) α (n ) α1 (nr ) α2 (n1 ) α2 (n2 ) α2 (nr ) a → P (a) = ... a(n1 , . . . , nr )x1p1 1 1 x2p1 1 2 . . . xrp x1p2 x2p2 . . . xrp .... 1 2 n1
nr
Here, the summation extends over all α (n1 ) α2 (n1 ) α3 (n1 ) p3 ..., p2
n1 = p1 1
α (n2 ) α2 (n2 ) α3 (n2 ) p2 p3 ...,
n2 = p1 1
... α (nr ) α2 (nr ) α3 (nr ) p2 p3 ...
nr = p1 1
in N. The correspondence is clearly one-to-one and onto from Ar (R) to the set R{. . . , xipj , . . . } = R{x1p1 , x1p2 , . . . }{x2p1 , x2p2 , . . . } . . . {xrp1 , xrp2 , . . . } of all such power series. Moreover, addition is preserved, and, as we will see below, P (f ∗ g) = P (f )P (g). Thus the ring Ar (R) is isomorphic to the ring R{. . . , xipj , . . . } of all r-fold formal power series. We now check that P (f ∗ g) = P (f )P (g). P (f )P (g) α1 (n1 ) α1 (n2 ) α1 (nr ) α2 (n1 ) α2 (n2 ) α2 (nr ) ... f (n1 , . . . , nr )x1p1 x2p1 . . . xrp1 x1p2 x2p2 . . . xrp2 . . . = n1
n2
nr
α1 (m1 ) α1 (m2 ) α1 (mr ) α2 (m1 ) α2 (m2 ) α2 (mr ) . . . g(m1 , . . . , mr )x1p1 x2p1 . . . xrp1 x1p2 x2p2 . . . xrp2 . . . m1 m2
=
mr
f (n1 , . . . , nr )g(m1 , . . . , mr )
n1 ,...,nr m1 ,...,mr α (n1 )+α1 (m1 ) α1 (n2 )+α1 (m2 ) α1 (nr )+α1 (mr ) x2p1 . . . xrp 1
x1p1 1
α (n1 )+α2 (m1 ) α2 (n2 )+α2 (m2 ) α2 (nr )+α2 (mr ) x2p2 . . . xrp ... 2
x1p2 2
174 =
Alexandru Zaharescu and Mohammad Zaki
f (n1 , . . . , nr )g(m1 , . . . , mr )
n1 ,...,nr m1 ,...,mr α (n1 m1 ) α1 (n2 m2 ) α1 (nr mr ) α2 (n1 m1 ) α2 (n2 m2 ) α2 (nr mr ) x2p1 . . . xrp x1p2 x2p2 . . . xrp ... 1 2
x1p1 1 =
k1 ,...,kr
···
k1 =m1 n1
f (n1 , . . . , nr )g(m1 , . . . , mr )
kr =mr nr
α (k1 ) α1 (k2 ) α2 (kr ) α1 (kr ) α2 (k1 ) α2 (k2 ) x1p2 x2p2 . . . xrp ... x2p1 . . . xrp 1 2
x1p1 1 =
α (k1 ) α1 (k2 ) α1 (kr ) α2 (k1 ) α2 (k2 ) α2 (kr ) x2p1 . . . xrp x1p2 x2p2 . . . xrp ... 1 2
f ∗ g(k1 , . . . , kr )x1p1 1
k1 ,...,kr
= P (f ∗ g). Let us also remark that this isomorphism preserves the absolute value, in the sense that |a|t = |P (a)| t , for any a ∈ Ar (R).
5. Generating degrees Recall that for two commutative topological rings A ⊂ B, a subset M ⊂ B is said to be a generating set of B over A if the ring A[M] is dense in B. The generating degree of B/A, gdeg(B/A) ∈ N ∪ {0, ∞} is defined by gdeg(B/A) : = min{|M|,
where M is a generating set of B/A}.
|M| denotes the number of elements of M if M is finite and ∞ if M is not finite. Let R be an integral domain, and F be its field of fractions. Let r ≥ 1 be an integer, and p be a prime number. Then Ar (R) is embedded in AR (F ). Let k be an integer with 1 ≤ k ≤ r. Let Br,k,p (R) denote the set of all f ∈ Ar (R) satisfying the condition that for all n1 , . . . , nk , nk+1 , . . . , nr ∈ N for which p divides nk one has f (n1 , . . . , nr ) = 0. Lemma 3. For each r, k, p, and R as above, Br,k,p (R) is a R-subalgebra of f ∈ Ar (R). Moreover, for any t = (t1 , . . . , tr ), Br,k,p (R) is topologically closed in Ar (R). Proof. Denote as usual ker θk,p = {f ∈ Ar (R): θk,p f = 0}. We observe that Br,k,p (R) = ker θk,p . From general theory it follows that ker θk,p , and hence also Br,k,p (R), is an Rsubalgebra of Ar (R). Moreover, θk,p is continuous with respect to each absolute value of the form ||t on Ar (R) by Theorem 1. It follows that ker θk,p is topologically closed in Ar (R). Hence Br,k,p (R) is closed in Ar (R), and the lemma is proved. Theorem 2. For any integral domain R, any integer r ≥ 1, any k ∈ {1, . . . , r}, any prime number p, and any t = (t1 , . . . , tr ) with t1 , . . . , tr > 0, t1 , . . . , tr linearly independent over Q, one has gdeg(Ar (R)/Br,k,p (R)) = 1, where gdeg is computed with respect to the topology on Ar (R) defined by the absolute value ||t .
Ring of arithmetical functions
175
Proof. Since by Lemma 3 the ring Br,k,p (R) is topologically closed in Ar (R) but it does not coincide with Ar (R), it follows that Br,k,p (R) is not dense in Ar (R). We deduce that gdeg Ar (R)/Br,k,p (R) ≥ 1. On the other hand, by employing the topological isomorphism from §4, we derive from Lemma 2 that Ar (R) has a generating element over Br,k,p (R). This means that gdeg Ar (R)/Br,k,p (R) ≤ 1. Combining the above inequalities we obtain gdeg(Ar (R)/Br,k,p (R)) = 1, which completes the proof of the theorem.
References [1] Alexandru V, Popescu N and Zaharescu A, On the closed subfields of Cp , J. Number Theory 68(2) (1998) 131–150 [2] Alexandru V, Popescu N and Zaharescu A, The generating degree of Cp , Canad. Math. Bull. 44(1) (2001) 3–11 [3] Alkan E, Zaharescu A and Zaki M, Arithmetical functions in several variables, Int. J. Number Theory 1(3) (2005) 383–399 [4] Alkan E, Zaharescu A and Zaki M, Unitary convolution for arithmetical functions in several variables, Hiroshima Math. J. 36(1) (2006) 113–124 [5] Cashwell E D and Everett C J, The ring of number-theoretic functions, Pacific J. Math. 9 (1959) 975–985 [6] Schwab E D, M¨obius categories as reduced standard division categories of combinatorial inverse monoids, Semigroup Forum 69(1) (2004) 30–40 [7] Schwab E D, The M¨obius category of some combinatorial inverse semigroups, Semigroup Forum 69(1) (2004) 41–50 [8] Schwab E D and Silberberg G, A note on some discrete valuation rings of arithmetical functions, Arch. Math. (Brno) 36 (2000) 103–109 [9] Schwab E D and Silberberg G, The valuated ring of the arithmetical functions as a power series ring, Arch. Math. (Brno) 37 (2001) 77–80 [10] Yokom K L, Totally multiplicative functions in regular convolution rings, Canadian Math. Bull. 16 (1973) 119–128
