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 [2]. As an immediate consequence we deduce that if V$(g_1,..., g_r)$ $\subseteq 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