Trivial extensions defined by Prüfer conditions

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. © 2009 Elsevier B.V. All rights reserved.