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 a class of examples of faithfully flat Ohm–Rush algebras with the McCoy property that are not weak content algebras. 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 fidel (A) property), we show that it is equivalent that \(R/I \rightarrow S/IS\) is McCoy for all ideals.