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/M is 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.

Key Words:

• Content ideal
• Gaussian polynomials
• Pullbacks
• Trivial ring extensions

Mathematics Subject Classification:

• Primary 13Agg
• Secondary 13D05, 13B02

ACKNOWLEDGMENT

We would like to thank the referee for his/her useful suggestions and comments, which have greatly improved this paper.

Notes

Communicated by I. Swanson.