The McCoy property in Ohm-Rush algebras

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.