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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This article deals mainly with rings (with zerodivisors) in which regular Gaussian polynomials have locally principal contents. Precisely, we show that if T M is a local ring which is not a field, D is a subring of T/M such that qf D = T/M, h T → T/Mis the canonical surjection and R = h −1 D , then if T satisfies the property "every regular Gaussian polynomial has locally principal content," then also R verifies the same property. We also show that if D is a Prüfer domain and T satisfies the property "every Gaussian polynomial has locally principal content", then R satisfies the same property. The article includes a brief discussion of the scopes and limits of our result.

In this paper we introduce and investigate a class of those rings in which every principal ideal is finitely presented. We establish the transfer of this notion to the trivial ring extension, direct product and homomorphic image, and then generate new and original families of rings satisfying this property.