The McCoy property in Ohm-Rush algebras

TL;DR

This paper investigates the McCoy property in Ohm-Rush algebras, providing examples and characterizations, especially under conditions like faithful flatness and Noetherianity, to deepen understanding of content and zero-divisors.

Contribution

It offers the first example of a faithful flat Ohm-Rush algebra with the McCoy property that is not a weak content algebra and characterizes when such algebras are weak content.

Findings

Provided an example of a faithful flat Ohm-Rush algebra with McCoy property not being a weak content algebra.

Showed that faithful flatness and McCoy property imply the algebra is weak content iff certain quotient maps are McCoy.

Established equivalences for Noetherian rings relating McCoy property in quotient maps to the algebra's content properties.

Abstract

An Ohm-Rush algebra $R \rightarrow S$ is called *McCoy* if for any zero-divisor $f$ in $S$, its content $c(f)$ has nonzero annihilator in $R$, because McCoy proved this when $S=R[x]$. We answer a question of Nasehpour by giving an example of a faithfully flat Ohm-Rush algebra with the McCoy property that is not a weak content algebra. However, we show that a faithfully flat Ohm-Rush algebra is a weak content algebra iff $R/I \rightarrow S/I S$ is McCoy for all radical (resp. prime) ideals $I$ of $R$. When $R$ is Noetherian (or has the more general \emph{fidel (A)} property), we show that it is equivalent that $R/I \rightarrow S/IS$ is McCoy for all ideals.