SOME RESULTS ON THE NONVANISHING OF CERTAIN EXT MODULES

Abstract

Let $R$ be a commutative Noetherian ring, $a$ an element of $R$, and $M$ a nonzero finitely generated $R$-module. We study the nonvanishing of ${\text{Ext }}_R^1(R∕Ra,M)$ and we give some conditions which guarantee that this is the case. In particular, we show that over a Noetherian local ring $(R,\mathfrak{𝔪})$, for an ideal $\mathfrak{𝔟}$ of $R$, if $\mu (\mathfrak{𝔪}∕\mathfrak{𝔟})\le \text{dim}(R∕\mathfrak{𝔟})+1$, then for each $a\in \mathfrak{𝔪}\setminus {\bigcup }_{\mathfrak{𝔭}\in {\text{mAss}}_R(R∕\mathfrak{𝔟})}\mathfrak{𝔭}$, we have ${\text{Ext}}_R^1(R∕Ra,R∕\mathfrak{𝔟})\ne 0$.