Gaussian Polynomials and Content Ideal in Pullbacks

Abstract

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.

Transfer of Prufer-like conditions

“…3) If R 1 ×R 2 is an arithmetical ring, then, for each i = 1, 2, R i is an arithmetical ring as homomorphic image of an arithmetical ring by Theorem 3.1 (2).…”

On Prüfer-like conditions

“…(1) The homomorphic image of a Gaussian ring is Gaussian. (2) The homomorphic image of an arithmetical ring is arithmetical.…”

“…(1) R is semihereditary, i.e., every finitely generated ideal of R is projective. (2) The weak global dimension of R is at most one.…”

Armendariz and SFT Properties in Subring Retracts

“…And a domain is Gaussian if and only if it is a Prüfer domain. See for instance [1], [5], [9], [11]. b i X i ∈ R[X] satisfy f g = 0, we have C(f )C(g) = 0 (that is a i b j = 0 for every i and j).…”