Chain conditions on composite Hurwitz series rings

Abstract In this paper, we study chain conditions on composite Hurwitz series rings and composite Hurwitz polynomial rings. More precisely, we characterize when composite Hurwitz series rings and composite Hurwitz polynomial rings are Noetherian, S-Noetherian or satisfy the ascending chain condition on principal ideals.


Hurwitz series rings
The formal power series rings and polynomial rings have been of interest and have had important applications in many areas, one of which has been differential algebra. In [1], Keigher introduced a variant of the ring of formal power series and studied some of its properties. In [2], Keigher called such a ring the ring of Hurwitz series and examined its ring theoretic properties. Since then, many works on the ring of Hurwitz series have been done ( [3][4][5]).
Let R be a commutative ring with identity, ROEOEX (resp., ROEX ) the formal power series ring (resp., polynomial ring) over R, and H.R/ the set of formal expressions of the form P 1 nD0 a n X n , where a n 2 R. Define addition and -product on H.R/ as follows: for f D P 1 nD0 a n X n ; g D P 1 nD0 b n X n 2 H.R/, .a n C b n /X n and f g D 1 X nD0 c n X n ; where c n D P n kD0 n k a k b n k and n k D nŠ .n k/ŠkŠ for nonnegative integers n k. Then H.R/ becomes a commutative ring with identity containing R under these two operations, i.e., H.R/ D .ROEOEX ; C; /. The ring H.R/ is called the Hurwitz series ring over R. The Hurwitz polynomial ring h.R/ over R is the subring of H.R/ consisting of formal expressions of the form P n kD0 a k X k , i.e., h.R/ D .ROEX ; C; /. Let R Â D be an extension of commutative rings with identity, and let H.R; D/ D ff 2 H.D/ j the constant term of f belongs to Rg (resp., h.R; D/ D ff 2 h.D/ j the constant term of f belongs to Rg). Then H.R; D/ (resp., h.R; D/) is a commutative ring with identity. We call H.R; D/ (resp., h.R; D/) a composite Hurwitz series ring (resp., composite Hurwitz polynomial ring). More precisely, H.R; D/ (resp., h.R; D/) is a subring of H.D/ (resp., h.D/) containing H.R/ (resp., h.R/), i.e., H.R; D/ D .R C XDOEOEX ; C; / (resp., h.R; D/ D .R C XDOEX ; C; /), where R C XDOEOEX D ff 2 DOEOEX j the constant term of f belongs to Rg (resp., R C XDOEX D ff 2 DOEX j the constant term of f belongs to Rg). Hence if R¨D, then H.R; D/ (resp., h.R; D/) gives algebraic properties of Hurwitz series (resp., Hurwitz polynomial) type rings strictly between two Hurwitz series rings (resp., Hurwitz polynomial rings). Also, it is easy to see that H.R; D/ (resp., h.R; D/) is a pullback of R and H.D/ (resp., h.D/).

Noetherian rings and related rings
Chain conditions have for many years been important tools in commutative algebra and algebraic geometry because of their use in producing many theorems and applications. For example, a relation between the ascending chain conditions on ideals and finitely generatedness of ideals in rings permits an interesting measure of the size and behavior of such rings, and the Noetherian condition plays a significant role to prove many results on varieties, homology and cohomology. Recently, Anderson and Dumitrescu [6] introduced the notion of S-Noetherian rings and gave a number of S-variants of well-known results for Noetherian rings. After them, S-Noetherian rings have been studied by some mathematicians (see [7][8][9][10][11]).
In [10,[12][13][14], the authors characterized when composite rings R C XDOEOEX and R C XDOEX are Noetherian rings, S-Noetherian rings, or satisfy the ascending chain condition on principal ideals. It was shown that RCXDOEOEX (resp., R C XDOEX ) is a Noetherian ring if and only if R is a Noetherian ring and D is a finitely generated R-module [13,Theorem 4]  . Also, they proved that for an anti-Archimedean subset S of R with zero characteristic containing an element s 0 2 S divisible in R by all the nonzero positive integers, if R is an S-Noetherian ring, then h.R/ is an S-Noetherian ring [4,Theorem 9.4]; and if R is an S -Noetherian ring and S consists of nonzerodivisors, then H.R/ is an S-Noetherian ring [4,Theorem 9.6].
In this paper, we study chain conditions on composite Hurwitz series rings H.R; D/ and composite Hurwitz polynomial rings h.R; D/, where R Â D is an extension of commutative rings with identity. In Section 2, we give necessary and sufficient conditions for the rings H.R; D/ and h.R; D/ to be Noetherian rings. We show that if char.R/ D 0, then H.R; D/ is a Noetherian ring if and only if h.R; D/ is a Noetherian ring, if and only R is a Noetherian ring and D is a finitely generated R-module containing Q. In Section 3, we give equivalent conditions for the rings H.R; D/ and h.R; D/ to be S-Noetherian rings, where S is an anti-Archimedean subset of R. We show that if char.R/ D 0 and S is an anti-Archimedean subset of R consisting of nonzerodivisors of D which contains an element divisible in D by all the positive integers, then H.R; D/ is an S-Noetherian ring if and only if R is an S-Noetherian ring and D is an S-finite R-module; and if char.R/ D 0 and S is an anti-Archimedean subset of R which contains an element divisible in D by all the positive integers, then h.R; D/ is an S -Noetherian ring if and only if R is an S-Noetherian ring and D is an S-finite R-module. In Section 4, we study when the rings H.R; D/ and h.R; D/ are présimplifiable. We prove that H.R; D/ is présimplifiable if and only if Z.D/ \ R Â 1 C U.R/, where Z.D/ is the set of zero-divisors of D and U.R/ is the set of units in R. We also prove that if D is a torsionfree Z-module, then h.R; D/ is présimplifiable if and only if D is a domainlike ring. Finally, in Section 5, we characterize when the rings H.R; D/ and h.R; D/ satisfy the ascending chain condition on principal ideals. We show that if D is a présimplifiable ring, then H.R; D/ satisfies the ascending chain condition on principal ideals if and only if U.D/ \ R D U.R/ and for each sequence .d n / n 1 of D with the property that for each n 1, there exists an element r n 2 R such that d n D d nC1 r n , d 1 D Â d 2 D Â is stationary; and if D is a présimplifiable ring with char.D/ > 0, then h.R; D/ satisfies the ascending chain condition on principal ideals if and only if U.D/ \ R D U.R/ and for each sequence .d n / n 1 of D with the property that for each n 1, there exists an element r n 2 R such that d n D d nC1 r n , d 1 D Â d 2 D Â is stationary.

Noetherian rings
Let R be a commutative ring with identity. Then the mapping W ROEOEX ! H.R/ (resp., W ROEX ! h.R/) defined by P 1 nD0 a n X n D P 1 nD0 nŠa n X n .resp:; (1) D contains Q.
(2) The Z-module D is divisible and torsion-free.
(3) The mapping W R C XDOEOEX ! H.R; D/ defined by P 1 nD0 a n X n D P 1 nD0 nŠa n X n is a ring isomorphism.
Let R Â D be an extension of commutative rings with identity, and set XDOEOEX D ff 2 H.R; D/ j the constant term of f is zerog (resp., XDOEX D ff 2 h.R; D/ j the constant term of f is zerog). Then it is easy to see that XDOEOEX (resp., XDOEX ) is an H.R; D/-module (resp., h.R; D/-module).
We are now ready to study when composite Hurwitz rings H.R; D/ and h.R; D/ are Noetherian rings.
Theorem 2.2. Let R Â D be an extension of commutative rings with identity. If char.R/ D 0, then the following statements are equivalent.
(2) R is a Noetherian ring and D is a finitely generated R-module containing Q.
Since .X; X 2 ; : : : / is finitely generated, there exists a positive integer n such that X p n 2 .X; X 2 ; : : : ; X p n 1 /; so we can find suitable elements g 1 ; : : : ; g p n 1 2 H.R; D/ (resp., g 1 ; : : : ; g p n 1 2 h.R; D/) such that X p n D X g 1 C C X p n 1 g p n 1 . Comparing the coefficients of X p n in both sides, we get for some b 1 ; : : : ; b p n 1 2 D. Note that p divides p n k for all k D 1; : : : ; p n 1 [4, Lemma 7.3]; so p is a unit in D. Since all the prime numbers are units in D, all the nonzero integers are also units in D. Therefore D contains Q, and hence by Lemma 2.1, R C XDOEOEX (resp., R C XDOEX ) is a Noetherian ring. Thus R is a Noetherian ring and D is a finitely generated R-module [ (2) ) (1) Assume that R is a Noetherian ring and D is a finitely generated R-module. Then R C XDOEOEX (resp.,  The proof for the composite Hurwitz polynomial ring case is the same as that for the composite Hurwitz series ring case. When R D D in Theorem 2.2, we obtain (1) R is a Noetherian ring containing Q.
We next show that in Theorem 2.2, the condition that char.R/ D 0 is essential. Proof. Suppose on the contrary that E is a Noetherian ring. Then .X; X 2 ; : : : / is a finitely generated ideal of E; so there exists a positive integer q such that .X; X 2 ; : : : / D .X; X 2 ; : : : ; X q /. Let char.R/ D p k 1 1 p k m m , where p 1 ; : : : ; p m are distinct prime numbers. Then we can take a positive integer n such that p n i > q for all i D 1; : : : ; m; so X p n i 2 .X; X 2 ; : : : ; X p n i 1 / for all i D 1; : : : ; m. Therefore for each i D 1; : : : ; m, there exist suitable elements g i1 ; : : : ; g i.p n i 1/ 2 E such that X p n i D X g i1 C C X p n i 1 g i.p n i 1/ . By comparing the coefficients of X p n i in both sides, we get Note that p i divides

S -Noetherian rings
Let R be a commutative ring with identity, S a (not necessarily saturated) multiplicative subset of R, and M a unitary R-module. Recall from [6, Definition 1] that an ideal I of R is S -finite if there exist an element s 2 S and a finitely generated ideal J of R such that sI Â J Â I ; and R is an S -Noetherian ring if each ideal of R is S -finite. Also, we say that the R-module M is S-finite if sM Â F Â M for some s 2 S and some finitely generated R-module F ; and M is S-Noetherian if each R-submodule of M is S-finite.
Our first result in this section gives a necessary condition for composite Hurwitz rings H.R; D/ and h.R; D/ to be S-Noetherian rings, where R Â D is an extension of commutative rings with identity and S is a multiplicative subset of R. Proposition 3.1. Let R Â D be an extension of commutative rings with identity, S be a .not necessarily saturated/ multiplicative subset of R, and E be either H.R; D/ or h.R; D/. If E is an S-Noetherian ring, then the following assertions hold.
(1) S contains an element s divisible in D by all the prime numbers.
Proof. (1) Suppose that E is an S -Noetherian ring. Then .X; X 2 ; : : : / is S-finite; so there exist s 2 S and f 1 ; : : : ; f m 2 .X; X 2 ; : : : / such that s .X; X 2 ; : : : / Â .f 1 ; : : : ; f m /. Let p be any prime number. Since f 1 ; : : : ; f m 2 .X; X 2 ; : : : /, we can find a positive integer n such that s .X; X 2 ; : : : / Â .X; X 2 ; : : : ; X p n 1 /. Therefore sX p n 2 .X; X 2 ; : : : ; X p n 1 /, and hence we can write sX p n D X g 1 C C X p n 1 g p n 1 for some g 1 ; : : : ; g p n 1 2 E. Comparing the coefficients of X p n in both sides, we obtain Let R be a commutative ring with identity and S a (not necessarily saturated) multiplicative subset of R. We say that S is anti-Archimedean if T n 1 s n R \ S ¤ ; for every s 2 S. We also say that an integral domain R is an anti-Archimedean domain if T n 1 a n R ¤ 0 for each 0 ¤ a 2 R (see [16]). Thus R is an anti-Archimedean domain if and only if R n f0g is an anti-Archimedean subset of R. Clearly, every multiplicative subset consisting of units is anti-Archimedean. Also, if V is a valuation domain with no height-one prime ideal (or equivalently, every nonzero prime ideal of V has infinite height), then V n f0g is an anti-Archimedean subset of V [16, Proposition 2.1].
We next characterize when a composite Hurwitz series ring H.R; D/ is an S-Noetherian ring under the assumption that S is an anti-Archimedean subset of R.    We end this section with an example satisfying the conditions in Theorems 3.2 and 3.3. More precisely, we construct an integral domain R, not containing Q, with char.R/ D 0 such that there exists an anti-Archimedean subset S of R containing an element divisible in R by all the positive integers.
Example 3.6. Let Z be the ring of integers and G the weak direct sum of fZ i g 1 iD1 which has the reverse lexicographic order, where Z i D Z for all positive integers i . Let fX i g 1 iD1 [ fY i g 1 iD1 be a set of indeterminates over Q and v be the valuation on Q fX i g 1 i D1 ; fY i g 1 i D1 induced by the mapping X i 7 ! 0 and Y i 7 ! e i , of where e i is an element of G whose i-th component is 1 and j -th component is 0 for j ¤ i .  Note that H.R/ (resp., h.R/) is not isomorphic to ROEOEX (resp., ROEX ) because R does not contain Q. Since R D R n f0g is anti-Archimedean subset of R and R is R -Noetherian, it follows from [4, Theorem 9.6] (resp., [4,Theorem 9.4]) that H.R/ (resp., h.R/) is an R -Noetherian ring.

Présimplifiable rings
Let R be a commutative ring with identity, U.R/ the set of units of R, and Z.R/ the set of zero-divisors of R. Recall that R is présimplifiable if whenever a; b 2 R satisfy ab D a, either a D 0 or b 2 U.R/. It was shown in [18] that R is présimplifiable if and only if Z.R/ Â 1 C U.R/. In [4], the authors studied when Hurwitz rings H.R/ and h.R/ are présimplifiable. In this section, we modify some properties of elements (units and nilpotent) of H.R/ and h.R/ in [4] to give equivalent conditions for composite Hurwitz series rings and composite Hurwitz polynomial rings to be présimplifiable. Our first result in this section is a necessary and sufficient condition for a composite Hurwitz series ring H.R; D/ to be présimplifiable, where R Â D is an extension of commutative rings with identity.  We next study when a composite Hurwitz polynomial ring h.R; D/ is présimplifiable, where R Â D is an extension of commutative rings with identity. To do this, we need two lemmas. (1) f is nilpotent if and only if a 0 is nilpotent and for each i D 1; : : : ; n, a i is nilpotent or some power of a i is with torsion.
(2) f is a unit if and only if a 0 is a unit in R and for each i D 1; : : : ; n, a i is nilpotent or some power of a i is with torsion. Proof.
(2) ()) Assume that f is a unit in h.R; D/. Then we can find an element g D P m iD0 b i X i 2 h.R; D/ such that f g D 1; so a 0 b 0 D 1. Hence a 0 is a unit in R. Since h.R; D/ Â h.D/, f is a unit in h.D/; so for each i D 1; : : : ; n, a i is nilpotent or some power of a i is with torsion [4, Theorem 3.1].
(() Assume that for each i D 1; : : : ; n, a i is nilpotent or some power of a i is with torsion. Then by (1), P n i D1 a i X i is nilpotent in h.R; D/. Since a 0 is a unit in h.R; D/, f is a unit in h.R; D/. (2) D is a torsion-free Z-module.
Proof. (1) ) (2) Let a be any nonzero element in D. If there exists a positive integer n such that na D 0, then 0 D naX n D aX X n 1 ; so X n 1 is a zero-divisor of h.R; D/. By the assumption, we can find an element d 2 D n f0g such that dX n 1 D 0, which is absurd. Thus D is a torsion-free Z-module. Let R be a commutative ring with identity. Recall that R is a domainlike ring if every zero-divisor of R is nilpotent.
It is easy to see that R is domainlike if and only if .0/ is primary. We next show that in Proposition 4.5, the condition that D is a torsion-free Z-module is essential.

Rings satisfying ascending chain condition on principal ideals
Let R be a commutative ring with identity. We say that R satisfies the ascending chain condition on principal ideals (ACCP) if there does not exist a strict ascending chain of principal ideals of R. It was shown in [19,Theorem 2.4] that if R Â D is an extension of integral domains with char.D/ D 0, then H.R; D/ satisfies ACCP if and only if h.R; D/ satisfies ACCP, if and only if T n 1 r 1 r n D D .0/ for each infinite sequence .r n / n 1 consisting of nonzero nonunits of R. In this section, we study an equivalent condition for H.R; D/ and h.R; D/ to satisfy ACCP, where R Â D is an extension of présimplifiable rings with identity.
Theorem 5.1. Let R Â D be an extension of commutative rings with identity. If D is a présimplifiable ring, then the following statements are equivalent. be an ascending chain of nonzero principal ideals of H.R; D/. Then for each n 1, f n D f nC1 g n for some g n 2 H.R; D/. If f n is a unit for some n 1, then there is nothing to prove; so we assume that f n is a nonunit for all n 1. For each n 1, write f n D P 1 mDk n a nm X m and g n D P 1 mD0 b nm X m , where a nk n ¤ 0. Since f n is a multiple of f nC1 , k 1 k 2 0; so there exists a positive integer q such that k n D k q for all n q. Hence a nk n D a nC1k nC1 b n0 for all n q. By the assumption, the chain a qk q D Â a qC1k qC1 D Â is stationary; so we can find an integer p q such that a mk m D D a pk p D for all m p. Therefore for each n p, there exists an element d n 2 D such that a nC1k nC1 D a nk n d n . Hence a nk n D a nk n d n b n0 for all n p. Since D is présimplifiable and a nk n ¤ 0, d n b n0 is a unit in D, which indicates that b n0 2 U.D/ \ R D U.R/. Hence g n is a unit in H.R; D/ [19, Lemma 2.2(1)], which shows that f n H.R; D/ D f p H.R; D/ for all n p. Thus H.R; D/ satisfies ACCP.
Let R Â D be an extension of commutative rings with identity. Note that by Lemma 4.3(2), if char.R/ > 0, then P n iD0 a i X i 2 h.R; D/ is a unit if and only if a 0 is a unit in R. Hence a similar argument as in the proof of Theorem 5.1 shows the following result.
Theorem 5.2. Let R Â D be an extension of commutative rings with identity. If D is a présimplifiable ring with char.D/ > 0, then the following statements are equivalent.
(2) U.D/ \ R D U.R/ and for each sequence .d n / n 1 of D with the property that for each n 1, there exists an element r n 2 R such that d n D d nC1 r n , d 1 D Â d 2 D Â is stationary.
When R D D in Theorems 5.1 and 5.2, we obtain Corollary 5.3. Let R be a présimplifiable ring with identity. Then the following assertions hold.
We are closing this paper with an example which shows that if a ring has characteristic zero, then ACCP property does not ascend into the Hurwitz polynomial ring extension.
Example 5.4. Let K be a field with char.K/ D 0, fA n g 1 nD1 a set of indeterminates over K, and set S D KOEfA n g 1 nD1 =.fA nC1 .A n A nC1 /g 1 nD1 /. Let a n be the image of A n in S and R be the localization of S at the ideal .a 1 ; a 2 ; : : : /S. (1) R is a présimplifiable ring which satisfies ACCP ( [12,Remark 4.17] and [20,Example]).
(2) Note that for all n 1, a n X C 1 D .a nC1 X C 1/ ..a n a nC1 /X C 1/; so .a 1 X C 1/ h.R/ Â .a 2 X C 1/ h.R/ Â is an ascending chain of principal ideals of h.R/. Suppose on the contrary that the chain stops. Then there exists a positive integer m such that a mC1 X C 1 D .a m X C 1/ f for some f 2 h.R/. Now, an easy calculation shows that f D P 1 nD0 b n X n , where b 0 D 1 and b n D . 1/ nC1 nŠa n 1 m .a mC1 a m / for all n 1. Since char.K/ D 0, b n ¤ 0 for all nonnegative integers n [20,Example]. Hence f 6 2 h.R/, which is absurd. Thus h.R/ does not satisfy ACCP.