In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three classes of modules give us a new characterization of the first modules, and confirm that there is an analogy between the notion of "Gorenstein projective, injective, and flat modules" and the notion of the usual "projective, injective, and flat modules".

In this paper, we prove that the global Gorenstein projective dimension of a ring R is equal to the global Gorenstein injective dimension of R, and that the global Gorenstein flat dimension of R is smaller than the common value of the terms of this equality. D. BENNIS AND N. MAHDOU 2. Proofs of the main results

This paper investigates coherent-like conditions and related properties that a trivial extension R := A ∝ E might inherit from the ring A for some classes of modules E. It captures previous results dealing primarily with coherence, and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition. INTRODUCTIONAll rings considered in this paper are commutative with identity elements and all modules are unital. Let A be a ring and E an A-module. The trivial ring extension of A by E is the ring R := A ∝ E whose underlying group is A × E with multiplication given by (a, e)(a ′ , e ′ ) = (aa ′ , ae ′ + a ′ e). Considerable work, part of it summarized in Glaz's book [20] and Huckaba's book (where R is called the idealization of E in A) [21], has been concerned with trivial ring extensions. These have proven to be useful in solving many open problems and conjectures for various contexts in (commutative and non-commutative) ring theory.A ring R is coherent if every finitely generated ideal of R is finitely presented; equivalently, if (0 : a) and I ∩ J are finitely generated for every a ∈ R and any two finitely generated ideals I and J of R [20]. Examples of coherent rings are Noetherian rings, Boolean algebras, von Neumann regular rings, valuation rings, and Prüfer/semihereditary rings. The concept of coherence first sprang up from the study of coherent sheaves in algebraic geometry, and then developed, under the influence of Noetherian ring theory and homology, towards a full-fledged topic in algebra. During the past 30 years, several (commutative) coherentlike notions grew out of coherence such as finite conductor, quasi-coherent, v-coherent, ncoherent, and -to some extent-GCD and G-GCD rings (see the respective definitions in the beginning of Sections 2 and 3). Noteworthy is that both the ring-theoretic and homological aspects of coherence run through most of these generalizations (see for instance [19]). This paper investigates coherent-like conditions and related properties that a trivial extension R := A ∝ E might inherit from the ring A for some classes of modules E. It captures previous results dealing primarily with coherence [20,30], and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions [17] (see also [7,13,11,12,14]). Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition.The second section provides a ring-theoretic approach. We first extend the definition of a v-coherent domain to rings with zerodivisors and develop a theory of these rings parallel to Glaz's study of finite conductor, quasi-coherent, and G-GCD rings [19]. Afterwards, we study the possible transfer of all these notions for various trivial extension contexts. Thereby, new examples are provided which, particularly, enrich the current literature with new classes of coherent-l...

Communicated by A.V. Geramita MSC: 13F05 13B05 13A15 13D05 13B25 a b s t r a c t This paper deals with well-known extensions of the Prüfer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zero-divisors subject to various Prüfer conditions. The new examples give further evidence for the validity of the Bazzoni-Glaz conjecture on the weak global dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.

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