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Trivial extensions defined by Prüfer conditions. (English) Zbl 1175.13008

Summary: 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.

MSC:

13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13B05 Galois theory and commutative ring extensions
13A15 Ideals and multiplicative ideal theory in commutative rings
13D05 Homological dimension and commutative rings
13B25 Polynomials over commutative rings
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References:

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