Let $R$ be a regular local ring and $M$ an $R$-module (not necessarily finite), then the injective dimension $\operatorname{id}_R(M)$ of $M$ is finite. When $M$ is finitely generated, we have $\dim(M)\leq \operatorname{id}_R(M)$.
When $M$ is an arbitrary $R$-module, does the relation $\dim(M)\leq \operatorname{id}_R(M)$ still hold?
(Here $\dim(M)$ denotes the supremum of the lengths of all chains of prime ideals in $\operatorname{Supp}(M)$.)