Polynomial Extensions of Commutative Rings

Abstract

This paper aims to generalize results on single variable polynomial rings over commutative rings with zerodivisors to the case of polynomial rings in arbitrarily many variables. Given a commutative ring R, we give necessary and sufficient conditions for the ring of polynomials with coefficients in R in arbitrarily many variables to be a PVMR and Krull ring. In answering these questions, we make use of the t and v operations on ideals as a means of characterizing these rings. We also give conjectures on necessary and sufficient conditions for an arbitrary polynomial ring to be a Dedekind ring, a UFR, and integrally closed.