Abstract
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.
Highlights
All rings considered in this paper are commutative with identity elements and all modules are unital
“Prufer domains have assumed a central role in the development of multiplicative ideal theory through numeral equivalent forms
This paper investigates the transfer of the above-mentioned Prufer conditions to trivial ring extensions
Summary
All rings considered in this paper are commutative with identity elements and all modules are unital. A problem initially associated with Kaplansky and his student Tsang [1, 2, 14, 22, 26] and termed as Tsang-Glaz-Vasconcelos conjecture in [16] sustained that “every nonzero Gaussian polynomial over a domain has an invertible (or, equivalently, locally principal) content ideal.”. This was made possible by the main result which states that a trivial ring extension of a domain by its quotient field satisfies the condition that “every Gaussian polynomial has locally principal content ideal.”. The section closes with a discussion -backed with examples- which attempts to rationalize this statement
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