A CHARACTERIZATION OF GORENSTEIN DEDEKIND DOMAINS

In this paper, we show that a domain R is a Gorenstein Dedekind domain if and only if every divisible module is Gorenstein injective; if and only if every divisible module is copure injective. Mathematics Subject Classification (2010): 16E05, 16E10


Introduction
Throughout this paper, all rings are commutative rings with identity element and all modules are unitary.For an R-module M , pd R M (resp.id R M , resp.fd R M ) stands for the projective (resp.injective, resp.flat) dimension of M .We also use w.gl.dim(R) (resp.gl.dim(R)) to denote the weak global (resp.global) dimension of R.
An R-module D is said to be divisible if Ext 1 R (R/aR, D) = 0 for all a ∈ R; and an R-module M is called h-divisible if it is an epic image of an injective R-module.
Note that injective modules and all h-divisible R-modules are divisible.
Divisible modules and h-divisible modules play important roles in characterizing domains.It is well known that a domain R is a Dedekind (resp.Prüfer) domain if and only if every divisible module is injective (resp.FP-injective); if and only if every h-divisible module is injective (resp.FP-injective).
Recall that a domain R is called a Matlis domain [9] if the projective dimension of the field of quotients is at most one.It is shown [10] Recall from [11] that an R-module W is called weak-injective if Ext 1 R (M, W ) = 0 for all modules M with fd R M ≤ 1 and from [1] that a domain R is called almost perfect (APD shortly) if all its proper homomorphic images are perfect.It is proved in [8,Corollary 6.4.8] that a domain R is an APD if and only if every divisible module is weak-injective; if and only if every h-divisible is weak-injective.
An R-module M is said to be Gorenstein projective (G-projective for short) [5] if there is an exact sequence of projective modules such that M ∼ = Im(P 0 → P 0 ) and that Hom R (−, Q) leaves the sequence P exact whenever Q is a projective R-module.A Gorenstein injective R-module is defined dually.The Gorenstein projective, injective dimensions are defined in terms of Gorenstein projective, injective resolutions, respectively, and denoted by Gpd R (−), Gid R (−).In [3], Bennis and Mahdou defined the Gorenstein global dimension Ggldim(R) of R, and proved that for any ring R, we have Recall that a ring R is called Gorenstein hereditary if Ggldim(R) ≤ 1.Also, a Gorenstein hereditary domain is called a Gorenstein Dedekind domain.Naturally, we propose the following question: Question 1.1.Let R be a domain.Is it true that R is a Gorenstein Dedekind domain if and only if every divisible module is Gorenstein injective; if and only if every h-divisible module is Gorenstein injective?
As in [4], Enochs and Jenda introduce the concepts of copure injective modules and strongly copure injective modules.For an R-module M , M is called copure injective if Ext 1 R (E, M ) = 0 for any injective R-module E, and M is called strongly copure injective if Ext i R (E, M ) = 0 for any injective R-module E and for all i ≥ 1.In the paper [4] the authors define the copure injective dimension cid R M of an In this paper, in terms of copure injective modules, we show that a domain R with ciD(R) ≤ 1 is exactly a Gorenstein Dedekind domain, and give an affirmative answer to Question 1.1.

Main result
Lemma 2.1.Let R be a ring with ciD(R) ≤ 1.Then every copure injective Rmodule M is divisible.Moreover, if R is a domain with ciD(R) ≤ 1, then every divisible R-module is copure injective.
Proof.Let M be a copure injective R-module.For any a ∈ R which is neither a non-zero-divisor nor a unit, fd R R/aR ≤ 1 and the sequence 0 X be an R-module.Note that (aR) + ∼ = R + as R-modules.Then we can obtain fd R (R/aR) + ≤ 1 from the sequence 0 = Tor R 3 (X, (aR)  Proof.The assertion follows from the fact that pd R E ≤ 1 holds for any injective R-module E by [7].
Let M be an R-module.As in [7], the copure projective dimension cpd R (M ) of an R-module M is defined to be the smallest integer n ≥ 0 such that Ext n+i R (M, F ) = 0

R
-module M to be the largest integer n ≥ 0 such that Ext n R (E, M ) = 0 for some injective R-module E. Of course, if no such n exists, write cid R (M ) = ∞.Thus cid R M = 0 if and only if M is strongly copure injective.As in [4, Lemma 3.1], it is shown that for an R-module M , cid R M ≤ m if and only if Ext m+i R (E, M ) = 0 for A CHARACTERIZATION OF GORENSTEIN DEDEKIND DOMAINS 99 any injective R-module E. The copure injective dimension of a ring R is defined in [7] as ciD(R) = sup{ cid R (M ) | M is an R-module }.It is clear that all domains R with ciD(R) ≤ 1 are Matlis domains.

Lemma 2 . 3 .
Let R be a domain.Then ciD(R) ≤ 1 if and only if every h-divisible module is copure injective.
that a domain R is a