A criterion for regular sequences

Let $R$ be a commutative noetherian ring and $f_{1}, ..., f_{r} \in R$. In this article we give (cf. the Theorem in \S2) a criterion for $f_{1}, ..., f_{r}$ to be regular sequence for a finitely generated module over $R$ which strengthens and generalises a result in \cite{2}. As an immediate consequence we deduce that if ${\rm V}(g_{1}, ..., g_{r}) \subseteq {\rm V} (f_{1}, >..., f_{r})$ in Spec $R$ and if $f_{1}, ..., f_{r}$ is a regular sequence in $R$, then $g_{1}, ..., g_{r}$ is also a regular sequence in $R$.