The amalgamated duplication of a ring along a semidualizing ideal

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The Amalgamated Duplication of a Ring Along a Semidualizing Ideal

MARYAMSALIMI(*) - ELHAMTAVASOLI(**) - SIAMAKYASSEMI(***)

ABSTRACT- LetRbe a commutative Noetherian ring and letIbe an ideal ofR. In this paper, after recalling briefly the main properties of the amalgamated duplication ringRIwhich is introduced by D'Anna and Fontana, we restrict our attention to the study of the properties ofRI, whenIis a semidualizing ideal ofR, i.e.,I is an ideal ofRandIis a semidualizingR-module. In particular, it is shown that if Iis a semidualizing ideal andMis a finitely generatedR-module, thenMis totally I-reflexive as anR-module if and only ifMis totally reflexive as an (RI)-module.

In addition, it is shown that ifIis a semidualizing ideal, thenRandIare Gorenstein projective overRI, and every injectiveR-module is Gorenstein injective as an (RI)-module. Finally, it is proved that ifIis a non-zero flat ideal ofR, then fdR(M)fdRI(MR(RI))fdR(MR(RI)), for everyR-moduleM.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 13D05, 13H10.

KEYWORDS. Amalgamated duplication, semidualizing, totally reflexive, Gorenstein projective.

1. Introduction

Throughout this paper all rings are considered commutative with identity element, and all ring homomorphisms are unital. Over a commutative local ring, semidualizing modules provide a common generalization

(*) Indirizzo dell'A.: Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

E-mail: maryamsalimi@ipm.ir

(**) Indirizzo dell'A.: Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

E-mail: elhamtavasoli@ipm.ir

(***) Indirizzo dell'A.: Department of Mathematics, University of Tehran, and School of mathematics, Institute for research in fundamental sciences (IPM), Tehran, Iran.

E-mail: yassemi@ipm.ir

The research of Siamak Yassemi was in part supported by a grant from IPM (No. 91130214).

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of a dualizing (canonical) module and a free module of rank one. In [10], Golod introduced the notion of semidualizing modules, though suggestions of this notion can be traced back to Foxby [9] and Vasconcelos [18].

LetRbe a ring and letMbe anR-module. The idealizationR^jM(also called trivial extension) introduced by Nagata in 1956 [14, Page 2], is a new ring where the moduleMcan be viewed as an ideal such that its square is 0.

In [11, Lemma 3.1], it is proved that if C is a semidualizing module for Noetherian ring R, then every injective (R^j C)-module is a direct summand of theR-module Hom_R(R^j C;E), whereEis an injectiveR-module.

Also, Holm and Jùrgensen in [11, Lemma 3.3], showed that if C is a semidualizing module for Noetherian ringR, then

(i) RandCare Gorenstein projective overR^j C.

(ii) Hom_R(R;E)Eand Hom_R(C;E) are Gorenstein injective over R^j C.

This paper builds on work of Holm and Jùrgensen for the amalgamated duplication instead of idealization. First, we recall the definition of amalgamated duplication. In [6] D'Anna and Fontana, considered a different type of construction obtained involving a ringRand an idealIRthat is denoted byRI, called amalgamated duplication, and it is defined as the following subring ofRR:

RI f(r;ri)jr2R;i2Ig:

They showed thatRIis reduced if and only ifRis reduced. This construction has been studied by different authors, since it allows to produce rings satisfying preassigned conditions and this is the main motivation for the investigations of the authors. In this paper, we study the ringRIfor semidualizing ideal I. In Section 3, we show that if Iis an ideal of Noe- therian ringRsuch thatIis a semidualizingR-module, andMis a finitely generatedR-module, thenM is totallyI-reflexive as anR-module if and only ifMis totally reflexive as an (RI)-module (see Theorem 3.2). Also, we show that, ifIis a non-zero ideal of Noetherian ringRsuch thatIis a semidualizingR-module andMis a totally reflexive (RI)-module, then

Tor^R_i(M;N)0Extⁱ_R(M;I_RN)

for alli1, provided thatNis a locally finite flat dimensionR-module, and Extⁱ_R(M;G)0Tor^R_i(M;Hom_R(I;G))

for all i1, provided that G is a locally finite injective dimension R- module.

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Finally, we prove that ifIis an ideal of Noetherian ringR, then every injective (RI)-module is a direct summand of the R-module Hom_R(RI;M), whereMis an injectiveR-module. In addition, we show that ifIis semidualizing, thenRandIare Gorenstein projective overRI, and every injectiveR-module is Gorenstein injective as an (RI)-module.

2. Preliminaries

First, we deal with some applications of a general construction, introduced in [6], called amalgamated duplication of a ring along an ideal.

LetRbe a commutative ring with unit element 1 and letIbe an ideal of R. Set

RI f(r;s)jr;s2R;s r2Ig:

It is easy to check thatRIis a subring, with unit element (1;1), ofRR (with the usual componentwise operations) and that RI f(r;ri)jr2R;i2Ig. In the following, we recall some main properties of the ringRIfrom [5] which we use in the next section.

PROPOSITION2.1. LetRbe a ring and letIbe an ideal ofR. Then the following statements hold.

(i) By introducing a multiplicative structure in the R-module direct sum RI by setting

(r;i)(s;j)(rs;rjsiij);

the map f :RI!RI defined by f((r;i))(r;ri) is a ring isomorphism and R-isomorphism too. Moreover, there is an split exact sequence of R-modules

0!R!^W RI!^c I!0

whereW(r)(r;r)for all r2R, andc((r;s))s r, for all(r;s)2RI.

Notice that this sequence splits.

(ii) R and RI have the same Krull dimension. Also, if R is a Noetherian ring, then RI is a finitely generated R-module.

In [1], [5], [6], [16], and [17], the properties of the ring RI were studied extensively.

The following notion was introduced by Golod, though suggestion of this notion can be traced back to Foxby and Vasconcelos.

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DEFINITION2.2. Let C be a finitely generated R-module. An R-module G is totally C-reflexive if it satisfies the following conditions:

(i) G is finitely generated over R;

(ii) the biduality map d^C_G:G !HomR(HomR(G;C);C) given by d^C_G(g)(c)c(g)for each g2G andc2Hom_R(G;C), is an isomorphism; and

(iii) Extⁱ_R(G;C)0Extⁱ_R(Hom_R(G;C);C)for all i1.

In this paper, for anR-moduleM, we use ``Mis totally reflexive'' instead of ``Mis totallyR-reflexive''.

The following notion was introduced, seemingly independently and using different terminology, by Foxby, Golod, Wakamatsu, and Vasconcelos.

DEFINITION2.3. The R-module C is semidualizing if it satisfies the following:

(i) C is finitely generated;

(ii) the homothety mapx_C^R:R !Hom_R(C;C)given byx_C^R(r)(c)rc for each r2R and c2C, is an isomorphism; and

(iii) Extⁱ_R(C;C)0for all i1.

In addition an R-module D is called dualizing if it is semidualizing andidR(D)51.

By the term ``Iis a semidualizing ideal for ringR'', we mean thatIis an ideal ofRandIis a semidualizingR-module.

Recall that a complete projective resolution is an exact sequence of projective modules

P: !P1 !P0 !P⁰ !P¹ ! ;

such that Hom_R(P;Q) is exact for every projective R-module Q. An R- moduleM is called Gorenstein projective, if there exists a complete projective resolutionPwithMcoker(P0 !P⁰):The notion of Gorenstein injective modules are defined dually.

3. Main Results

Throughout this section, R is a Noetherian ring. We follow standard notation and terminology from [2].

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In this section, we study the ringRIfor a semidualizing idealI. In the following Theorem, we show that a finitely generatedR-moduleMis totallyI-reflexive if and only ifMis totally reflexive as an (RI)-module, provided thatIis a semidualizingR-module. First, we prove the following lemma:

LEMMA 3.1. Let I be a semidualizing ideal of R. Then the following statements hold.

(i) There is an(RI)-isomorphismHom_R(RI;I)RI.

(ii) There is a natural equivalence of R-modules Hom_R_I( ;RI)Hom_R( ;I):

(iii) Let M be an R-module. Then the biduality map M !Hom_R(Hom_R(M;I);I)

is an R-isomorphism if and only if the biduality map M !Hom_R_I(Hom_R_I(M;RI);RI) is an(RI)-isomorphism.

(iv) There is an(RI)-isomorphismHomRI(R;RI)I.

(v) If E is an injective R-module, then there is an equivalence of (RI)-module

Hom_R_I(Hom_R(RI;E); )Hom_R(Hom_R(I;E); ):

PROOF. (i) We have the following isomorphisms ofR-modules:

Hom_R(RI;I)Hom_R(RI;I)Hom_R(I;I)Hom_R(R;I)

RIRI;

sinceIis semidualizing. The proof of [1, Theorem 3.3] implies that theR- isomorphism HomR(RI;I)RIis an (RI)-isomorphism.

(ii) L et M be an R-module. In the following sequence, the first isomorphism is from (i), the second isomorphism is from Hom-tensor adjointness, and the third isomorphism is induced by tensor cancellation:

HomRI(M;RI) HomRI(M;HomR(RI;I))

Hom_R(M_R_I(RI);I)

Hom_R(M;I):

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(iii) In the next sequence, the first isomorphism is from (i), (ii) and Hom-tensor adjointness, and the second isomorphism is induced by tensor cancellation:

Hom_R_I(Hom_R_I(M;RI);RI)Hom_R(Hom_R(M;I)_R_I(RI);I)

HomR(HomR(M;I);I):

It shows that there is an (RI)-isomorphism

MHomRI(HomRI(M;RI);RI) if and only if there is anR-isomorphism

MHom_R(Hom_R(M;I);I):

By [15, Proposition 5.4.1 (a)], the biduality map d^R_M ^I is an (RI)-isomorphism if and only if the biduality mapd^I_Mis anR-isomorphism.

(iv) It follows from (ii). Note thatIhas (RI)-module structure by Proposition 2.1 (i).

(v) In the following sequence, the first isomorphism follows from (i), the second isomorphism follows from [7, Thorem 3.2.11], and the third isomorphism is induced by Hom-tensor adjointness:

Hom_R_I(Hom_R(RI;E); ) Hom_R_I(Hom_R(Hom_R(RI;I);E); )

Hom_R_I(Hom_R(I;E)_R(RI); )

HomR(HomR(I;E); ):

p THEOREM 3.2. Let I be a semidualizing ideal of R, and let M be a finitely generated R-module. Then M is totally I-reflexive as an R-module if and only if M is totally reflexive as an(RI)-module.

PROOF. L et E be an injective resolution of I as an R-module. Let W:R!RIbeR-homomorphism in Proposition 2.1(i). By [15, Proposi- tions 2.1.12 and 2.1.13],IandRare totallyI-reflexive, so according to [15, Proposition 2.1.4],RIis totallyI-reflexive. Therefore,RIis totallyI- reflexive by Proposition 2.1(i). Hence Extⁱ_R(RI;I)0 for alli1, so the complex Hom_R(RI;E) is an injective resolution of the (RI)- module HomR(RI;I). By Lemma 3.1(i), HomR(RI;E) is an injective resolution of the (RI)-moduleRI. This yields the first isomorphism in the next sequence:

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Extⁱ_R_I(M;RI) H _i(Hom_R_I(M;Hom_R(RI;E)))

H _i(HomR((RI)RIM;E))

H _i(Hom_R(M;E))

Extⁱ_R(M;I):

()

It follows that Extⁱ_R_I(M;RI)0 for all i1 if and only if Extⁱ_R(M;I)0 for alli1. Also, we have the following isomorphisms:

Extⁱ_R_I(Hom_R_I(M;RI);RI) Extⁱ_R_I(Hom_R(M;I);RI)

Extⁱ_R(HomR(M;I);I):

In the above sequence, the first isomorphism follows from Lemma 3.1(ii) and the second one follows from (), using HomR(M;I) in place ofM. It follows that Extⁱ_R_I(Hom_R_I(M;RI);RI)0 for all i1 if and only if Extⁱ_R(Hom_R(M;I);I)0 for all i1. By Lemma 3.1(iii), the biduality mapM !Hom_R(Hom_R(M;I);I) is an isomorphism if and only if the biduality map M !Hom_R_I(Hom_RI(M;RI);RI) is an isomorphism. So, the assertion follows from definition. p COROLLARY3.3. Let I be a semidualizing ideal of R. Then I is totally reflexive as an(RI)-module.

PROOF. By [15, Proposition 2.1.12], I is totally I-reflexive as an R- module. So by Theorem 3.2,Iis totally reflexive as an (RI)-module. p COROLLARY3.4. Let I be a non-zero semidualizing ideal of R, and let M be a totally reflexive (RI)-module. Then the following statements hold.

(i) If N is a locally finite flat dimension R-module, then Tor^R_i(M;N)0Extⁱ_R(M;I_RN) for all i1.

(ii) If N is a locally finite injective dimension R-module, then Extⁱ_R(M;N)0Tor^R_i(M;HomR(I;N)) for all i1.

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PROOF. By Theorem 3.2, M is totally I-reflexive as an R-module.

Therefore, (i) follows from [15, Proposition 5.4.8], and (ii) follows from

[15, Proposition 5.4.9]. p

In [15], Sather-Wagstaff investigated the behavior of semidualizing and totally reflexive modules under flat based change of ring, indeed, in [15, Propositions 2.2.1 and 5.3.1], he showed that if W:R !S is a flat ring homomorphism, then the following statements hold.

(i) L etCbe a finitely generatedR-module. IfCis a semidualizing R-module, then CRSis a semidualizing S-module. The converse holds whenWis faithfully flat.

(ii) L et C be a semidualizing R-module and let M be a finitely generated R-module. If M is totally C-reflexive, then the S- moduleC_RMis totally (C_RS)-reflexive. The converse holds whenWis faithfully flat.

Assume that I is a non-zero flat ideal of Noetherian ring R. It is straightforward to see that the mapW:R !RIis faithfully flat, since RIRI, by Proposition 2.1(i). So we have:

LEMMA 3.5. Let I be a non-zero flat ideal of R. Then the following statements hold:

(i) Let C be a finitely generated R-module. Then C is semidualizing R-module if and only if CR(RI) is a semidualizing(RI)-module.

(ii) Let C be a semidualizing R-module and let M be a finitely generated R-module. Then M is totally C-reflexive if and only if the(RI)-module M_R(RI)is totally(C_R(RI))-reflexive.

PROPOSITION3.6. Assume thatIis a non-zero flat ideal ofR. LetCbe a semidualizingR-module, and letM be a totallyC-reflexiveR-module.

Then the following statements hold.

(i) If N is a locally finite flat dimension(RI)-module, then Tor^R_i ^I(M;N)0Extⁱ_R_I(MR(RI);CRN) for all i1.

(ii) If N is a locally finite injective dimension(RI)-module, then Tor^R_i ^I(M;HomRI(CR(RI);N))0Extⁱ_R_I(MR(RI);N)

for all i1.

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PROOF. Note thatRIis a flatR-module, sinceIis a flat ideal of R. By Lemma 3.5,CR(RI) is a semidualizing (RI)-module and M_R(RI) is a totally (C_R(RI))-reflexive (RI)-module.

(i) In the following sequence, the equality follows from [15, Proposition 5.4.8], and the isomorphism holds, sinceRIis a flatR-module

0 Tor^R_i ^I(MR(RI);N)

Tor^R_i ^I(M;N)_R(RI);

for all i1. So, Tor^R_i ^I(M;N)0, since RI is faithfully flat R- module.

In the following sequence, for all i1, the equality follows from [15, Proposition 5.4.8], and the isomorphism induces by tensor cancellation:

0 Extⁱ_R_I(MR(RI);(CR(RI))RIN)

Extⁱ_R_I(M_R(RI);C_RN):

(ii) In the following sequence, for all i1, the equality holds by [15, Proposition 5.4.9], and the isomorphism holds, since RIis a flat R-module

0 Tor^R_i ^I(MR(RI);HomRI(CR(RI);N))

Tor^R_i ^I(M;HomRI(CR(RI);N))R(RI):

So, Tor^R_i ^I(M;Hom_R_I(C_R(RI);N))0, since RI is a faithfully flat R-module. By [15, Proposition 5.4.9], we have

Extⁱ_R_I(M_R(RI);N)0:

p LetIbe a semidualizing ideal ofR. In the following, we investigate that R andIare Gorenstein projective overRI. Also, we show that every injectiveR-module is Gorenstein injective as an (RI)-module. First, we prove the following lemma:

LEMMA3.7. Let I be an ideal of R. Then the following statements hold.

(i) If M is a(faithfully)injective R-module, thenHom_R(RI;M) is a(faithfully)injective(RI)-module.

(ii) Every injective (RI)-module is a direct summand of the R-module HomR(RI;M), where M is an injective R-module.

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PROOF. (i) The following sequence of (RI)-isomorphisms makes clear that ifMis a (faithfully) injectiveR-module, then HomR(RI;M) is a (faithfully) injective (RI)-module

Hom_R_I( ;Hom_R(RI;M)) Hom_R((RI)_R_I ;M)

HomR( ;M):

Note that in the above sequence, the first isomorphism follows from Hom- tensor adjointness, and the second isomorphism is induced by tensor cancellation.

(ii) L etJbe an injective (RI)-module. It is enough to show thatJis embeded into anR-module of the form Hom_R(RI;M) where M is an injectiveR-module. ConsiderJ as anR-module and embed it into an injective R-module M. Then use isomorphisms in part (i), to convert the monomorphism ofR-modulesJ,!Mto a monomorphism of (RI)-mod-

ulesJ,!Hom_R(RI;M). p

THEOREM3.8. Let I be a semidualizing ideal of R, and let E be an injective R-module. Then the following statements hold.

(i) R and I are Gorenstein projective over RI.

(ii) HomR(R;E)E andHomR(I;E)are Gorenstein injective over RI.

PROOF. (i) By Lemma 3.1 (iv), Hom_R_I(R;RI)I. It means, the dual ofRwith respect to the ringRIisI. But dualization with respect to the ring preserves the class of finitely generated Gorenstein projective modules by [3, Observation (1.1.7)]. So, to prove part (i), it is enough to show that Ris Gorenstein projective overRI. By [4, Proposition 2.16], it is enough to show thatRis a totally reflexive (RI)-module, sinceRis a Noetherian ring andRis a finitely generated (RI)-module. By Theorem 3.2, it suffices to show thatR is a totallyI-reflexive R-module, and this follows from [15, Proposition 2.1.13].

(ii) First, we show that HomR(I;E) is Gorenstein injective overRI.

SinceIis Gorenstein projective overRI, it has a complete projective resolution as follows:

P: !P₁ ^@!¹ P₀ ^@!⁰ P ₁ ! ;

such that Pi are projective (RI)-modules and Icoker@¹, and HomRI(P;Q) is exact where Q is a projective (RI)-module. We can assume thatPconsists of finitely generated (RI)-modules, sinceRis a

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Noetherian ring. By Lemma 3.7(i), the (RI)-moduleJHomR(RI;E) is injective. Consider the exact complex KHomRI(P;J) of injective (RI)-modules. Note that for every injective (RI)-moduleLwe have

Hom_R_I(L;K) Hom_R_I(L;Hom_R_I(P;J))

Hom_R_I(L_R_IP;J)

HomRI(P;HomRI(L;J)):

()

Also, HomRI(L;J) is a flat (RI)-module, so it is the direct limit of projective (RI)-modulesQa, that is, Hom_R_I(L;J)lim! Qa:So by ()

Hom_R_I(L;K)Hom_R_I(P;lim! Qa)lim! Hom_R_I(P;Qa);

is exact. This shows thatKis a complete injective resolution overRIand ker (HomRI(@¹;J)) HomRI(coker(@¹);J)

Hom_R_I(I;Hom_R(RI;E))

Hom_R(I_R_I(RI);E)

HomR(I;E):

Therefore, Hom_R(I;E) is a Gorenstein injective (RI)-module.

Using the same argument, one can show that Hom_R(R;E)E is

Gorenstein injective overRI. p

In [12] and [13], some properties of homological dimension of RI were investigated.

PROPOSITION3.9. LetIbe a non-zero semidualizing ideal of local ring R. ThenIis a dualizingR-module if and only ifRIis Gorenstein.

PROOF. By [13, Lemma 2.1], id_R_I(RI)id_R(I). So, the assertion

follows by definition. p

COROLLARY3.10. Let I be a non-zero semidualizing ideal of local ring R. Then I is an injective R-module if and only if RI is a quasi- Frobenius ring.

PROOF. Recall that the ringRis called quasi-Frobenius ifRis a Gor- enstein ring of Krull dimension zero (see [7]). The assertion follows from

Proposition 3.9. p

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PROPOSITION3.11. Let I be a non-zero flat ideal ofR. For every R- moduleM we have:

(i) id_R(M)id_R(M_R(RI)) (ii) fdR(M)fdR(MR(RI))

PROOF. Note thatRIis a faithfully flatR-module. (i) follows from [20, Corollary 2.9] and (ii) follows from [20, Corollary 2.11]. p COROLLARY3.12. Let I be a non-zero flat ideal of R. For every R-module M, we have

fd_R(M)fd_R_I(M_R(RI))fd_R(M_R(RI)):

PROOF. By [12, Lemma 2.3], we have fd_R(M)fd_R_I(M_R(RI)), and by Proposition 3.11, fdR(M)fdR(MR(RI)): p PROPOSITION 3.13. Let I be a non-zero ideal of a ring R of Krull dimension zero. If RI is a Gorenstein ring, then id_R(RI) fd_R(RI).

PROOF. Note that dim(RI)dim(R)0, soRIis self-injective.

By [20, Corollary 3.4], id_R(RI)fd_R(RI). p THEOREM3.14. Let I be an ideal of Artinian ring R. Then R is Gor- enstein if and only ifExtⁱ_R(k;R)0.

PROOF. By assumption, R is Gorenstein if and only if R is self injective. Note that Ris an Artinian ring if and only if Ris an Artinian (RI)-module. So, the assertion follows [8, Theorem 6.1]. p Acknowledgments. The authors are very grateful to the referee for several comments that greatly improved the paper.

REFERENCES

[1] A. BAGHERI- M. SALIMI- E. TAVASOLI- S. YASSEMI,A Construction of Quasi- Gorenstein Rings, J. Algebra Appl.11, no. 1 (2012), 1250013 (9 pages).

[2] W. BRUNS- J. HERZOG,Cohen-Macaulay Rings, Cambridge University press, Cambridge, 1993.

[3] L. W. CHRISTENSEN, Gorenstein Dimensions, Lecture Notes in Math. Vol.

1747, Springer, Berlin, 2000.

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[4] L. W. CHRISTENSEN - H. B. FOXBY - H. HOLM, Beyond Totally Reflexive Modules and Back: a survey on Gorenstein dimensions, ``Commutative Algebra-Noetherian and non-Noetherian Perspectives'', 101-143, Springer, New York, 2011.

[5] M. D'ANNA,A construction of Gorenstein rings, J. Algebra306 (2006), pp.

507-519.

[6] M. D'ANNA- M. FONTANA,An amalgamated duplication of a ring along an ideal, J. Algebra Appl.6, no. 3 (2007), pp. 443-459.

[7] E. E. ENOCHS- O. M. G. JENDA,Relative Homological Algebra, de Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter and Co., Berlin, 2000.

[8] N. EPSTEIN- Y. YAO, Criteria for Flatness and Injectivity, Math. Z.271, no. 3-4 (2012), pp. 1193-1210.

[9] H. B. FOXBY, Gorenstein Modules and Related Modules, Math. Scand. 31, (1972), pp. 267-284.

[10] E. S. GOLOD,G-dimension and Generalized Perfect Ideals, Trudy Mat. Inst.

Steklov.165, (1984), pp. 62-66.

[11] H. HOLM- P. JéRGENSEN,Cohen-Macaulay Homological Dimensions, Rend.

Semin. Mat. Univ. Padova,117(2007), pp. 87-112.

[12] M. CHHITI - N. MAHDOU, Homological dimension of the amalgamated duplication of a ring along a pure ideal, Afr. Diaspora J. Math. (N. S.)10, no. 1, (2010), pp. 1-6.

[13] N. MAHDOU- M. TAMEKKANTE,Gorenstein global dimension of an amalgamated duplication of a coherent ring along an ideal, Mediterr. J. Math.8, no. 3 (2011), pp. 293-305.

[14] M. NAGATA,Local Rings, Interscience, New York, 1962.

[15] S. SATHER-WAGSTAFF, Semidualizing modules, URL:http://www.ndsu.edu/

pubweb/ssatherw/.

[16] J. SHAPIRO,On a construction of Gorenstein rings proposed by M. D'Anna, J.

Algebra,323(2010), pp. 1155-1158.

[17] E. TAVASOLI- M. SALIMI- A. TEHRANIAN,Amalgamated duplication of some special rings, Bulletin of Korean Math. Soc.49no. 5 (2012) pp. 989-996.

[18] W. V. VASCONCELOS, Divisor Theory in Module Categories, North-Holland Math. Stud., vol. 14, North-Holland Publishing Co., Amsterdam, (1974).

[19] T. WAKAMATSU,On modules with trivial self-extensions, J. Algebra,114, no. 1 (1988), pp. 106-114.

[20] S. YASSEMI, On flat and injective dimension, Italian Journal of Pure and Applied Mathematics, N.6(1999), pp. 33-41.

Manoscritto pervenuto in redazione il 26 Ottobre 2011.

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