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 \(R\).