Auslander Modules

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May 11, 2017 - AC] 11 May 2017. AUSLANDER MODULES. PEYMAN NASEHPOUR. Abstract. In this paper, we introduce the notion of Auslander and Z- ...
arXiv:1705.03980v1 [math.AC] 11 May 2017

AUSLANDER MODULES PEYMAN NASEHPOUR

Abstract. In this paper, we introduce the notion of Auslander and Z-Auslander modules, inspired from Auslander’s Zero-Divisor Conjecture and give some interesting results for these modules. We also introduce and investigate the concept of modules having property (Z), a concept that is based on a property of flat modules.

0. Introduction Auslander’s Zero-Divisor Conjecture (ZDC for short) in commutative algebra states that if R is a Noetherian local ring, M is an R-module of finite type and finite projective dimension, and r ∈ R is not a zero-divisor on M , then r is not a zero-divisor on R [4, p. 8]. On the other hand, it is easy to see that if M is a flat R-module and r ∈ R is a zero-divisor on M , then r is a zero-divisor on R (Proposition 2.2). These two facts are the main motivation of the present paper. In fact, if we denote the set of zero-divisors of a module M over a ring R by ZR (M ), the question arises when the inclusion ZR (R) ⊆ ZR (M ) holds. In §1, we call an R-module M with the property ZR (R) ⊆ ZR (M ) an Auslander module and give a couple of families of Auslander modules (See Definition 1.1 and Proposition 1.2). The main theme of the current paper is to see under what conditions if M is an Auslander Rmodule, then the S-module M ⊗R S is Auslander, where S is an R-algebra (Check Theorem 1.4 and Theorem 1.9). In §2, we define an R-module M to have property (Z), if ZR (M ) ⊆ ZR (R) (See Definition 2.1) and introduce some families of modules having property (Z) (Check Theorem 2.3 and Theorem 2.4). We also investigate those Auslander modules, which have property (Z), i.e. Z-Auslander modules (Check Definition 2.5, Proposition 2.8 and Theorem 2.11). In this paper, all rings are commutative with non-zero identities and all modules are unital. 1. Auslander Modules Auslander’s Zero-Divisor Conjecture (ZDC for short) in module theory states that if R is a Noetherian local ring, M is an R-module of finite type and finite 2010 Mathematics Subject Classification. 13A15, 13B25, 13F25. Key words and phrases. Auslander modules, Auslander’s Zero-Divisor Conjecture, content algebras, Property Z, Z-Auslander modules. 1

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projective dimension and r ∈ R is not a zero-divisor on M , then r is not a zerodivisor on R [4, p. 8]. This inspires us to give the following definition: Definition 1.1. We define an R-module M to be an Auslander module, if r ∈ R is not a zero-divisor on M , then r is not a zero-divisor on R, or equivalently if the following property holds: ZR (R) ⊆ ZR (M ). Proposition 1.2 (Some Families of Auslander Modules). Let M be an R-module. Then the following statements hold: (1) If R is a domain, then M is an Auslander R-module. (2) The R-module M is Auslander if and only if M/N is an Auslander Rmodule for each R-submodule N of M . (3) If M is a flat and content R-module such that for any s ∈ R, there is an x ∈ M such that c(x) = (s). Then M is an Auslander R-module. (4) If for any nonzero s ∈ R, there is an x ∈ M such that s · x 6= 0, i.e. Ann(M ) = (0), then HomR (M, M ) is an Auslander R-module. (5) If M is an Auslander R-module, then M ⊕ M ′ is an Auslander R-module for any R-module M ′ . In particular, if {Mi }i∈Λ is a family of R-modules and there L is an i ∈QΛ, say i0 , such that Mi0 is an Auslander R-module, then i∈Λ Mi and i∈Λ Mi are Auslander R-modules. Proof. The proof of (1) and (2) is easy and left to the reader. We prove the other assertions: (3): Let r ∈ ZR (R). By definition, there is a nonzero s ∈ R such that r · s = 0. Since in content modules c(x) = (0) if and only if x = 0 ([12, Statement 1.2]), by assumption, there is a nonzero x ∈ M such that c(x) = (s). This implies that r·c(x) = (0). But since M is flat and content, by [12, Theorem 1.5], r·c(x) = c(r·x). This implies that r ∈ ZR (M ). (4): Let r ∈ ZR (R). So there is a nonzero s ∈ R such that r · s = 0. Define fs : M −→ M by fs (x) = s · x. By assumption, fs is a nonzero element of HomR (M, M ). But rfs = 0. This means that r ∈ ZR (Hom(M, M )). (5): Let r ∈ ZR (R). Since M is an Auslander R-module, there is a nonzero m ∈ M such that rm = 0. But (m, 0) ∈ M ⊕ M ′ is nonzero, while r(m, 0) = (0, 0). This means that r ∈ ZR (M ⊕ M ′ ). Now from this,Lit is obvious Q that if one of the modules in the family {Mi }i∈Λ is Auslander, then i∈Λ Mi and i∈Λ Mi are both Auslander R-modules.  Proposition 1.3. Let M be an Auslander R-module and S a multiplicatively closed subset of R contained in R − ZR (M ). Then MS is an Auslander RS -module. Proof. Let ZR (R) ⊆ ZR (M ) and S be a multiplicatively closed subset of R such that S ⊆ R − ZR (M ). Take r1 /s1 ∈ ZRS (RS ). So there exists an r2 /s2 6= 0/1 such that (r1 · r2 )/(s1 · s2 ) = 0/1. Since S ⊆ R − ZR (R), we have r1 · r2 = 0, where r2 6= 0. But ZR (R) ⊆ ZR (M ), so r1 ∈ ZR (M ). Consequently, there is a nonzero m ∈ M such that r1 · m = 0. Since S ⊆ R − ZR (M ), m/1 is a nonzero element of MS . This point that r1 /s1 · m/1 = 0/1, causes r1 /s1 to be an element of ZRS (MS ) and the proof is complete.  Let us recall that an R-module M has property (A), if each finitely generated ideal I ⊆ ZR (M ) has a nonzero annihilator in M ([9, Definition 10]). Examples

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of modules having property (A) include modules having very few zero-divisors, in particular finitely generated modules over Noetherian rings ([9, Definition 6]). Homological aspects of modules having very few zero-divisors have been investigated in [10]. Theorem 1.4. Let M be an Auslander R-module, have property (A) and G a commutative, cancellative, and torsion-free monoid. Then M [G] is an Auslander R[G]-module. Proof. Let f ∈ ZR[G] (R[G]). By [9, Theorem 2], there is a nonzero element r ∈ R such that f · r = 0. This implies that c(f ) ⊆ ZR (R). But M is an Auslander R-module, so ZR (R) ⊆ ZR (M ), which implies that c(f ) ⊆ ZR (M ). On the other hand, M has property (A). Therefore, c(f ) has a nonzero annihilator in M . Hence, f ∈ ZR[G] (M [G]) and the proof is complete.  Note that a semimodule version of Theorem 1.4 has been given in Theorem 1.22 in [8]. The following lemma is a generalization of Theorem 5 in [3]: Lemma 1.5. Let R be a Noetherian ring, M a finitely generated R-module, f ∈ R[[X]], g ∈ M [[X]] − {0}, and f g = 0. Then there is a nonzero constant m ∈ M such that f · m = 0. Proof. Define c(g), the content of g, to be the R-submodule of M generated by its coefficients. If c(f )c(g) = (0), then choose a nonzero m ∈ c(g). Clearly f · m = 0. Otherwise, by Theorem 3.1 in [1], one can choose a positive integer k, such that c(f )c(f )k−1 c(g) = 0, where c(f )k−1 c(g) 6= 0. Now for any nonzero element m ∈ c(f )k−1 c(g), we have f · m = 0 and the proof is complete.  Theorem 1.6. Let R be a Noetherian ring and M an Auslander R-module. If M has property (A), then M [[X]] is an Auslander R[[X]]-module. Proof. Let f ∈ ZR[[X]] (R[[X]]). By Lemma 1.5, there is a nonzero element r ∈ R such that f · r = 0. This implies that c(f ) ⊆ ZR (R). But M is an Auslander R-module, so ZR (R) ⊆ ZR (M ), which implies that c(f ) ⊆ ZR (M ). On the other hand, M has property (A). Therefore, c(f ) has a nonzero annihilator in M and so f ∈ ZR[[X]] (M [[X]]) and the proof is complete.  Remark 1.7 (Ohm-Rush Algebras). Let us recall that if B is an R-algebra, the content of f ∈ B, denoted by c(f ), is defined to be the following ideal: \ c(f ) = {I ∈ Id(R) : f ∈ IB}, I

where by Id(R), we mean the set of all ideals of R ([12]). The R-algebra B is said to be an Ohm-Rush R-algebra, if f ∈ c(f )B, for all f ∈ B ([2, Definition 2.1]). If R is a ring and f = a0 + a1 X + · · · + an X n + · · · is an element of R[[X]], then Af is defined to be the ideal of R generated by the coefficients of f , i.e. Af = (a0 , a1 , . . . , an , . . .). One can easily check that if R is Noetherian, then Af = c(f ) and therefore, R[[X]] is an Ohm-Rush R-algebra.

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Now we go further to define McCoy algebras, though we don’t go through them deeply in this paper. McCoy semialgebras (and algebras) and their properties have been discussed in more details in author’s recent paper on zero-divisors of semimodules and semialgebras [8]. Definition 1.8. We say that B is a McCoy R-algebra, if B is an Ohm-Rush Ralgebra and f · g = 0 with g 6= 0 implies that there is a nonzero r ∈ R such that c(f ) · r = (0), for all f, g ∈ B. Any content algebra is a McCoy algebra ([12, Statement 6.1]). So, we have plenty of examples for McCoy algebras (Also, check [11]). Now we proceed to give the following general theorem on Auslander modules: Theorem 1.9. Let M be an Auslander R-module and B a faithfully flat and McCoy R-algebra. If M has property (A), then M ⊗R B is an Auslander B-module. Proof. Let f ∈ ZB (B). So there is a nonzero r ∈ R such that c(f ) · r = (0). This implies that c(f ) ⊆ ZR (R). But M is an Auslander R-module. Therefore, c(f ) ⊆ ZR (M ). Since c(f ) is a finitely generated ideal of R ([12, p. 3]) and M has property (A), there is a nonzero m ∈ M such that c(f ) · m = (0). This means that c(f ) ⊆ AnnR (m). Therefore, c(f )B ⊆ AnnR (m)B. Since any McCoy R-algebra is an Ohm-Rush algebra, we have that f ∈ c(f )B. Our claim is that AnnR (m)B = AnnB (1 ⊗ m) and here is the proof: Since 0 −→ R/ AnnR (m) −→ M is an R-exact sequence and B is a faithfully flat R-module, we have the following B-exact sequence: 0 −→ B/ AnnR (m)B −→ M ⊗R B, with AnnR (m)B = Ann(m ⊗R 1B ). This means that f ∈ ZB (M ⊗R B) and the proof is complete.  Corollary 1.10. Let M be an Auslander R-module and B a content R-algebra. If M has property (A), then M ⊗R B is an Auslander B-module. Proof. By definition of content algebras ([12, Section 6]), any content R-algebra is faithfully flat. Also, by [12, Statement 6.1], any content R-algebra is a McCoy R-algebra.  Question 1.11. Is there any faithfully flat McCoy algebra that is not a content algebra? 2. Modules Having Property (Z) Let M be an R-module. Similar to what we have seen in the previous section, it is also interesting to ask when the inclusion ZR (M ) ⊆ ZR (R) holds and what properties this condition enjoys. This is the base for the following definition: Definition 2.1. We define an R-module M to have property (Z), if r ∈ R is not a zero-divisor on R, then r is not a zero-divisor on M , or equivalently, if the following property holds: ZR (M ) ⊆ ZR (R). Proposition 2.2. Every flat module has property (Z).

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Proof. Let r ∈ / ZR (R). This means that the following sequence is exact: r·

0→R− →R But M is R-flat. So the following sequence is exact: r·

0→M − →M And this means that r ∈ / ZR (M ), Q.E.D.



The proof of the following proposition is quite similar to the proof of Proposition 1.4. Therefore, we just mention the proof briefly. Theorem 2.3. Let the ring R have property (A), the R-module M have property (Z), and G be a commutative, cancellative, and torsion-free monoid. Then R[G]module M [G] has property (Z). Proof. Let f ∈ ZR[G] (M [G]). By [9, Theorem 2], there is a nonzero m ∈ M such that c(f ) · m = 0, which means that c(f ) ⊆ ZR (M ). Since M has property (Z), c(f ) ⊆ ZR (R), and since R has property (A), f ∈ ZR[G] (R[G]) and the proof is complete.  Theorem 2.4. Let R be a Noetherian ring and the R-module M finitely generated such that it has property (Z). Then the R[[X]]-module M [[X]] has property (Z). Proof. Let f ∈ ZR[[X]] (M [[X]]). By Lemma 1.5, there is a nonzero element m ∈ M such that f · m = 0. By Remark 1.7, this implies that c(f ) ⊆ ZR (M ). But M has property (Z), so by definition, ZR (M ) ⊆ ZR (R), which implies that c(f ) ⊆ ZR (R). On the other hand, since every Noetherian ring has property (A) ([5, Theorem 82, p. 56]), c(f ) has a nonzero annihilator in R. This means that f ∈ ZR[[X]] (R[[X]]), Q.E.D.  We continue this section by investigating those modules that are Auslander modules and have property (Z). For brevity, we give a name to this family of modules: Definition 2.5. We define an R-module M to be Z-Auslander if the R-module M is Auslander and has property (Z), or equivalently, if the following property holds: ZR (R) = ZR (M ). Remark 2.6. Definitely Definition 2.5 is justifiable if there are examples of modules that are Auslander but have not property (Z) and also there are some modules that have property (Z) but are not Auslander. (1) Let R be a ring and S ⊆ R − ZR (R) a multiplicatively closed subset of R. Then it is easy to see that ZR (R) = ZR (RS ), i.e. RS is a Z-Auslander R-module. (2) Let D be a domain and M a D-module such that ZD (M ) 6= {0}. Clearly ZD (D) = {0} and therefore, M is Auslander, while M has not property (Z). For example if D is a domain that is not a field, then D has an ideal I such that I 6= (0) and I 6= D. It is clear that ZD (D/I) ⊇ I. (3) Let k be a field and consider the ideal I = (0)⊕k of the ring R = k ⊕k. It is easy to see that ZR (R) = ((0)⊕k)∪(k⊕(0)), while ZR (R/I) = (0)⊕k. This means that the R-module R/I has property (Z), while it is not Auslander.

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Proposition 2.7 (Some Families of Z-Auslander Modules). Let M be an R-module. Then the following statements hold: (1) If R is a domain and M is a flat R-module, then M is Z-Auslander. (2) If R is a von Neumann regular ring, then every Auslander module is ZAuslander. (3) If M is a flat and content R-module such that for any s ∈ R, there is an x ∈ M such that c(x) = (s). Then M is a Z-Auslander R-module. (4) If R is a Noetherian ring and M is a finitely generated flat R-module and for any nonzero s ∈ R, there is an x ∈ M such that s · x 6= 0. Then HomR (M, M ) is a Z-Auslander R-module. (5) If M is an Auslander R-module, and M and M ′ are both flat modules, then M ⊕M ′ is a Z-Auslander R-module. In particular, if {Mi }i∈Λ is a family of flat R-modules and L there is an i ∈ Λ, say i0 , such that Mi0 is an Auslander R-module, then i∈Λ Mi is a Z-Auslander R-module. (6) If R is a coherent ring and {Mi }i∈Λ is a family of flat R-modules and Q there is an i ∈ Λ, say i0 , such that Mi0 is an Auslander R-module, then i∈Λ Mi is a Z-Auslander R-module. Proof. By Proposition 2.2, every Auslander and flat module is Z-Auslander. By considering Proposition 1.2, the proof of assertions 1–3 is straightforward. The proof of assertion 4 is easy based on Theorem 7.10 in [7] that says that each finitely generated flat module over a local ring is free. Now if R is a Noetherian ring and M is a flat and finitely generated R-module, then M is a locally free Rmodule. This causes HomR (M, M ) to be also a locally free R-module and therefore, HomR (M, M ) is R-flat and by Proposition 2.2, a Z-Auslander module. The proof of the assertions 5 and 6 is also easy, if we note that the direct sum of flat modules is flat ([6, Proposition 4.2]) and if R is a coherent ring, then the direct product of flat modules is flat ([6, Theorem 4.47]).  Proposition 2.8. Let M be a Z-Auslander R-module and both the ring R and the module M have property (A). If G is a commutative, cancellative, and torsion-free monoid, then R[G]-module M [G] is Z-Auslander. Proof. By Theorem 1.4 and Theorem 2.3, the assertion holds.



Theorem 2.9. Let M be an Auslander and a flat R-module and B a faithfully flat and McCoy R-algebra. If M has property (A), then M ⊗R B is a Z-Auslander B-module. Proof. By Theorem 1.9, ZB (B) ⊆ ZB (M ⊗R B). On the other hand, since M is a flat R-module, by [6, Proposition 4.1], M ⊗R B is a flat B-module. This implies that ZB (M ⊗R B) ⊆ ZB (B), Q.E.D.  Proposition 2.10. Let R be a Noetherian ring and M a finitely generated Rmodule. If M is a Z-Auslander R-module, then M [[X]] is a Z-Auslander R[[X]]module. Proof. Since M is finite and R is Noetherian, M is also a Noetherian R-module. This means that both the ring R and the module M have property (A). Now by Theorem 1.6 and Theorem 2.4 the proof is complete. 

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Let us recall that an R-module M is said to have few zero-divisors of size n, if ZR (M ) is a finite union of n prime ideals p1 , . . . , pn of R such that pi * pj for all i 6= j [9, Definition 13]. We end the current paper by giving the following result on formal power series modules: Theorem 2.11. Let R be a Noetherian ring and M be a finitely generated Rmodule. Then the following statements hold: (1) M [[X]] has few zero-divisors of size n if and only if M has few zero-divisors of size n and property (A). (2) The R[[X]]-module M [[X]] is Z-Auslander if and only if the R-module M is Z-Auslander. Proof. (1): Since R is Noetherian, for any ideal I of R, we have that I[[X]] = I · R[[X]] ([7, p. 5]). Now by Dedekind-Mertens Lemma for formal power series rings [1, Theorem 3.1], the proof is similar to the proof of Theorem 14 in [9]. (2): This is immediate from Theorem 1.6 and Theorem 2.4.  Acknowledgements The author was partly supported by Department of Engineering Science at University of Tehran and wishes to thank Professor Winfried Bruns for his advice. References [1] N. Epstein and J. Shapiro, A Dedekind-Mertens theorem for power series rings, Proc. Amer. Math. Soc. 144 (2016), 917–924. [2] N. Epstein and J. Shapiro, The Ohm-Rush content function, J. Algebra Appl., 15, No. 1 (2016), 1650009 (14 pages). [3] D. E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc. 27 (1971), no. 3, 427–433. [4] M. Hochster, Topics in the homological theory of modules over commutative rings, CBMS Regional Conf. Ser. in Math., 24, Amer. Math. Soc, Providence, RI, 1975. [5] I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970. [6] T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, Berlin, 1999. [7] H. Matsumura, Commutative Ring Theory, Vol. 8., Cambridge University Press, Cambridge, 1989. [8] P. Nasehpour, On zero-divisors of semimodules and semialgebras, arXiv preprint (2017), arXiv:1702.00810. [9] P. Nasehpour, Zero-divisors of semigroup modules, Kyungpook Math. J., 51 (1) (2011), 37–42. [10] P. Nasehpour and Sh. Payrovi, Modules having few zero-divisors, Comm. Algebra 38 (2010), 3154–3162. [11] D. G. Northcott, A generalization of a theorem on the content of polynomials, Proc. Cambridge Phil. Soc. 55 (1959), 282–288. [12] J. Ohm and D. E. Rush, Content modules and algebras, Math. Scand. 31 (1972), 49–68. Peyman Nasehpour, Department of Engineering Science, University of Tehran, Tehran, Iran E-mail address: [email protected]