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1 Communications in Algebra ISSN: (Print) (Online) Journal homepage: n-coherent Rings Sang Bum Lee To cite this article: Sang Bum Lee (2002) n-coherent Rings, Communications in Algebra, 30:3, , DOI: / To link to this article: Published online: 26 Dec Submit your article to this ournal Article views: 85 View related articles Citing articles: 13 View citing articles Full Terms & Conditions of access and use can be found at Download by: [National Cheng Kung University] Date: 21 September 2017, At: 21:44

2 COMMUNICATIONS IN ALGEBRA, 30(3), (2002) n-coherent RINGS Sang Bum Lee Department of Mathematical Education, Sangmyung University, Seoul , Korea ABSTRACT The notion of left n-coherent ring (for integers n > 0) is introduced. n-coherent rings are characterized in various ways, using n-flat and n-absolutely pure modules. By a ring R we mean an associative ring with 1, and by a module a unital (left or right) R-module. A ring R is called left coherent if every finitely generated left ideal of R is finitely presented. Coherent rings have been characterized in various ways. The deepest result is the one due to Chase [1] which claims that the ring R is left coherent if and only if products of flat right R-modules are again flat if and only if products of copies of R are flat right R-modules. For other characterizations of coherency, see Stenstro m [9], and (in the commutative case) Glaz [4]. Here we wish to introduce a generalization of coherency: we shall call a ring R left n-coherent (for integers n > 0orn ¼1) if every finitely generated submodule of a free left R-module whose proective dimension is n 1is finitely presented. Accordingly, all rings are left 1-coherent, and the left coherent rings are exactly those which are d-coherent (d denotes the left global dimension of R). In particular, left 1-coherent rings are left coherent Copyright # 2002 by Marcel Dekker, Inc.

3 1120 LEE We wish to prove that left n-coherent rings have a number of characterizations, in particular in terms of generalizations of flatness and absolute purity. Recall that a right R-module M is flat if Tor R 1 ðm; NÞ ¼0 holds for all finitely presented left R-modules N. The following generalization of flatness will be used. A right R-module M will be called n-flat if Tor R 1 ðm; NÞ ¼0 holds for all finitely presented left R-modules N with p:d: R N n (p.d. ¼ proective dimension). Let us observe right away: Lemma 1. If R is a commutative domain, then an R-module M is 1-flat if and only if it is torsion-free. Proof. Let Q denote the field of quotients of R, and E an inective cogenerator of the category of R-modules. Suppose the R-module F is 1-flat, i.e. it satisfies Tor R 1 ðn; FÞ ¼0 for all finitely presented R-modules N of proective dimension 1. The natural isomorphism Ext 1 R ðn; Hom RðF; EÞÞ ffi Hom R ðtor R 1 ðn; FÞ; EÞ ð1þ implies that Hom R ðf; EÞ is divisible (see Fuchs-Salce [3, p.36]). For the torsion submodule tf we have the RD-exact sequence 0! tf! F! F=tF! 0 which induces the RD-exact sequence 0! Hom R ðf=tf; EÞ! Hom R ðf; EÞ!Hom R ðtf; EÞ!0. Here Hom R ðtf; EÞ is Hausdorff in the R- topology and (as we saw above) divisible, thus it is 0. Hence tf ¼ 0, and F is torsion-free. Conversely, if F is a torsion-free R-module, then the inection map F! Q R F induces an epimorphism Hom R ðq R F; EÞ!Hom R ðf; EÞ. The module Hom R ðq R F; EÞ is h-divisible, and therefore so is Hom R ðf; EÞ. Thus, by [3, 36], for an R-module N, p:d: R N 1 implies that Ext 1 R ðn; Hom RðF; EÞÞ ¼ 0, whence Hom R ðtor R 1 ðn; FÞ; EÞ ¼0 follows. By the choice of E, we conclude Tor R 1 ðn; FÞ ¼0; showing that F is 1-flat. We will say that a finitely generated left R-module N has a finiteproective resolution if it has a long proective resolution with finitely generated proective left R-modules. Note that a finitely presented left module N with p:d:n n over any left n-coherent ring R admits a finite-proective resolution. A left R-module A is said to be absolutely pure (or FP-inective)ifitisa pure submodule in every left R-module containing it as a submodule. A familiar result on absolute purity states:

4 n-coherent RINGS 1121 Lemma 2 ðmegibben ½7ŠÞ. A left R-module A is absolutely pure if and only if, for all finitely presented left R-modules N, it satisfies Ext 1 RðN; AÞ ¼0: Accordingly, we define a left R-module A to be n-absolutely pure if, for all finitely presented left R-modules N of proective dimension n, we have Ext 1 RðN; AÞ ¼0. Lemma 3. A module over a commutative domain R is 1-absolutely pure exactly if it is divisible. Proof. If D is a 1-absolutely pure R-module, then Ext 1 RðR=L; DÞ ¼0 for all proective ideals L. By Fuchs-Salce [3, p. 36], this amounts to the divisibility of D. Conversely, suppose D is a divisible module, and N is a finitely presented R-module of proective dimension 1. Since N has a finite-proective resolution, we can apply the natural isomorphism (see e.g. Rotman [8, p. 257]) Tor R 1 ðhom RðD; EÞ; NÞ ffihom R ðext 1 RðN; DÞ; EÞ ð2þ for an inective cogenerator E. Here Hom R ðd; EÞ is a torsion-free R-module, so by Lemma 1, it is 1-flat. This means that the Tor in the last formula vanishes, and consequently, the right side is 0. This leads to the equation Ext 1 RðN; DÞ ¼0, which amounts to the 1-absolute purity of D. We shall also need the following familiar result. Lemma 4 ðenochs ½2ŠÞ. Suppose N is a finitely generated left R-module. If Ext 1 RðN; AÞ ¼0 holds for all absolutely pure left R-modules A, then N is finitely presented. By the character module of the left (right) R-module M is meant the right (left) R-module M [ ¼ Hom Z ðm; Q=ZÞ. A well-known theorem of Lambek [5] states that M is a flat right R-module exactly if M [ is an inective left R-module. This generalizes to the following Lemma: Lemma 5. A right R-module B is n-flat if and only if its character module B [ is an n-absolutely pure left R-module. Proof. In view of the natural isomorphism Ext 1 R ðn; Hom ZðB; Q=ZÞÞ ffi Hom Z ðtor R 1 ðn; BÞ; Q=ZÞ

5 1122 LEE (where Q=Z is an inective cogenerator for abelian groups), it is clear that the right module B is n-flat if and only if B [ is an n-absolutely pure left module: ust choose N to be a finitely presented left R-module of proective dimension n. We shall also need the following two lemmas. Lemma 6 ðlambek ½6; p: 90ŠÞ. Every right (left) R-module M embeds as a submodule in the character module of a free left (right) R-module. Lemma 7 ðwarfield ½10ŠÞ. The canonical map M! M [[ embeds a left (right) R-module M in its double character module as a pure submodule. We start our discussion with the following result which is our main tool in the characterization of n-coherency in terms of n-absolute purity. Theorem 1. A ring R is left n-coherent if and only if, for every finitely presented left R-module N of proective dimension n, every n-absolutely pure left R-module A, satisfies: Ext 2 RðN; AÞ ¼0: Proof. Suppose R is a left n-coherent ring, and A is an n-absolutely pure left R-module. Let 0! H! F! N! 0 be a presentation of a finitely presented left R-module N with p:d:n n, where F is a finitely generated free R-module, and H is a finitely presented submodule with p:d:h n 1. In the induced exact sequence Ext 1 R ðh; AÞ!Ext2 R ðn; AÞ!Ext2 RðF; AÞ ¼0, the first Ext vanishes, thus the middle Ext is 0. Conversely, assume R is a ring such that the n-absolutely pure left R- modules A satisfy the stated criterion and I is a finitely generated left submodule of a finitely generated free left R-module F of proective dimension n 1. Then R=I is finitely presented with p:d:r=i n, so Ext 2 RðR=I; AÞ ¼ 0 by hypothesis. Hence Ext 1 RðI; AÞ ¼0 for all n-absolutely pure left R- modules A. By Lemma 4, I has to be finitely presented, i.e., R is left n- coherent. Note that if R is left n-coherent, then also Ext k RðN; AÞ ¼0 for all k 1 provided that A is an n-absolutely pure left and N is a finitely presented left R-module of proective dimension n. We continue with the following characterization pointing out another connection between n-coherency and n-absolute purity. Theorem 2. For a ring R, the following conditions are equivalent:

6 n-coherent RINGS 1123 ðiþ R is left n-coherent; ðiiþ quotients of n-absolutely pure left R-modules modulo pure submodules are n-absolutely pure; ðii Þ quotients of inective left R-modules modulo pure submodules are n-absolutely pure. Proof. (i) ) (ii) Assume R is left n-coherent, and B is a pure submodule of the n-absolutely pure left R-module A. For every finitely presented left R- module N of proective dimension n, we obtain the exact sequence 0 ¼ Ext 1 R ðn; AÞ!Ext1 R ðn; A=BÞ!Ext2 RðN; BÞ. The last Ext vanishes by Theorem 1, as B is n-absolutely pure. Thus Ext 1 RðN; A=BÞ ¼0, and A=B is n- absolutely pure. (ii) ) (ii 0 ) is trivial. (ii 0 ) ) (i) Let E be an inective left R-module containing the n-absolutely pure left R-module A. From the exact sequence 0! A! E! E=A! 0 we derive the exact sequence Ext 1 R ðn; E=AÞ!Ext2 RðN; AÞ! Ext 2 RðN; EÞ ¼0, where N denotes a finitely presented left R-module of proective dimension n. By (ii ), the first Ext is 0, thus Ext 2 RðN; AÞ ¼0, and R is left n-coherent by Theorem 1. Our next result points out how n-coherency is related to character modules; this generalizes results by Wisbauer [11]. Theorem 3. The following conditions on a ring R are equivalent: ðiþ R is left n-coherent; ðiiiþ a left R-module A is n-absolutely pure if and only if its character module A [ is an n-flat right R-module; ðivþ a left R-module A is n-absolutely pure if and only if its double character module A [[ is an n-absolutely pure (inective) left R- module. Proof. (i) ) (iii) For n ¼1, see Wisbauer [11]. Assume R is a left n- coherent ring, and A is a left R-module. We apply the analogue of (2) to a finitely presented module N of finite proective dimension (since it has a finite-proective resolution): Tor R 1 ðhom ZðA; Q=ZÞ; NÞ ffihom Z ðext 1 RðN; AÞ; Q=ZÞ: The left hand side vanishes for all finitely presented left R-modules N with p:d:n n exactly if Hom Z ða; Q=ZÞ ¼A [ is an n-flat right R-module. Q=Z being an inective cogenerator of the category of Z-modules, the right hand side vanishes for all finitely presented N with p:d:n n if and

7 1124 LEE only if Ext 1 RðN; AÞ ¼0 for all N. This is equivalent to the n-absolute purity of A. (iii) ) (i) We verify that (iii) implies (ii 0 ) in Theorem 2. Let E be an inective left R-module, and A a pure submodule of E. The pure-exact sequence 0! A! E! E=A! 0 induces the pure-exact sequence 0!ðE=AÞ [! E [! A [! 0. By hypothesis (iii), E [ is an n-flat right R- module. Evidently, the same is true for any pure submodule, in particular, for ðe=aþ [. Hence (iii) implies that E=A is n-absolutely pure, so by Theorem 2, R is left n-coherent. (iii), (iv) Apply Lemma 5 to B ¼ A [ to complete the proof. Additional characterizations are given in the next theorem. Theorem 4. The following conditions on a ring R are equivalent: ðiiiþ a left R-module A is n-absolutely pure if and only if its character module A [ is an n-flat right R-module; ðvþ for every inective left R-module E, the character module E [ is an n- flat right R-module; ðviþ a right R-module M is n-flat exactly if its double character module M [[ is an n-flat right R-module. Proof. (iii) ) (vi) Let M be an n-flat right R-module, i.e., M [ is an inective left R-module. By (iii), M [[ is an n-flat right R-module. That M [[ is n-flat implies M is n-flat is a simple consequence of Lemma 7. (vi) ) (v) Let E be an inective left R-module. By Lemma 6, E embeds as a submodule in, and hence is a summand of, the character module F [ of a free right R-module F. Hence E [ is a summand of F [[ which is n-flat by hypothesis. Therefore, E [ is an n-flat right R-module. (v) ) (iii) Assuming (v), suppose A is an n-absolutely pure left R- module. Let E be an inective left module containing A. The pure-exact sequence 0! A! E! E=A! 0 induces the pure-exact sequence 0!ðE=AÞ [! E [! A [! 0, whence the n-flatness of E [ implies the same for A [. To show that the n-flatness of A [ implies that A is n-absolutely pure, observe that by Lemma 5, A [[ is then an n-absolutely pure left module which contains, by Lemma 7, an isomorphic copy of A as a pure submodule. Thus A is n-absolutely pure. Recall that, over a domain R, flatness and torsion-freeness coincide if and only if R is a Pru fer domain (i.e. a semi-hereditary domain). Also, absolute purity and divisibility are equivalent exactly if R is Pru fer. This generalizes to the non-commutative case as follows.

8 n-coherent RINGS 1125 Corollary 1. The following conditions are equivalent for a ring R: ðiþ R is left semi-hereditary; ðiiþ 1-flat left R-modules are flat; ðiiiþ 1-absolutely pure left R-modules are absolutely pure. Proof. It suffices to note that all finitely presented left R-modules are of proective dimension 1 exactly if finitely generated submodules of finitely generated free left R-modules are proective. This is the case exactly if R is left semi-hereditary. Let us point out that Chase s characterization generalizes easily to our case, by using our characterization (iii). Indeed, we have: Theorem 5. For a ring R and for any n ð0 < n 1Þ, the following conditions are equivalent: (a) R is left n-coherent; (b) direct products of n-flat right R-modules are n-flat; (c) direct products of copies of R are right n-flat. Proof. (a) ) (b) Let M i ði 2 IÞ be a set of n-flat right R-modules, R left n- coherent. To show that the direct product M ¼ Q i2i M i is n-flat, we argue as follows. By Lemma 5, the modules M [ i are n-absolutely pure left R-modules, so their direct sum i2i M [ i is an absolutely pure left R-module. In view of Thoerem 3, the character module ð i2i M [ i Þ[ ffi Q i2i M[[ i is an n-flat right R- module. By Lemma 7, every module embeds in its double character module as a pure submodule, so M is isomorphic to a pure submodule of the n-flat right module Q i2i M[[ i, hence it is likewise n-flat. ðbþ )ðcþ is trivial: (c) ) (a) Let H be a finitely generated submodule of a finitely generated free left R-module F with p.d. R H 4 n 1, and I an arbitrary index set. By hypothesis, R I is an n-flat right R-module, thus Tor R 1(R I, F=H) ¼ 0. The proof given by Lenzing [7] carries over without change to conclude that H is finitely presented. Hence (a) follows, completing the proof. ACKNOWLEDGMENT The paper was completed while the author was visiting Tulane University. He expresses his thanks to the Department of Mathematics of Tulane University for the hospitality and Professor Laszlo Fuchs for several discussions on the subect.

9 1126 LEE REFERENCES 1. Chase, S. Direct Products of Modules. Trans. Amer. Math. Soc. 1960, 97, Enochs, E.E. A Note on Absolutely Pure Modules. Canad. Math. Bull. 1976, 19, Fuchs, L.; Salce, L. Modules Over Valuation Domains, Lecture Notes in Pure and Applied Math., vol. 97 Marcel Dekker: New York, Glaz, S. Commutative Coherent Rings, Lecture Notes in Mathematics, Springer: Berlin, 1989; Vol Lambek, J. A Module is Flat if and only if its Character Module is Inective. Canad. Math. Bull. 1964, 7, , Lectures on Rings and Modules; Blaisdell Publ. Co.: Waltham, Megibben, C. Absolutely Pure Modules. Proc. Amer. Math. Soc. 1970, 26, Rotman, J.J. An introduction to Homological Algebra; Academic Press: New York, Stenstro m, B. Coherent Rings and FP-inective Modules. J. London Math. Soc. 1970, 2, Warfield, R.B., Jr. Purity and Algebraic Compactness for Modules. Pac. J. Math., 1969, 28, Wisbauer, R. Foundation of Module and Ring Theory; Gordon and Breach Science Publ.: Philadelphia, Received May 2000 Revised June 2000

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