This might be a trivial question to experts but not to me whatsoever. Suppose that $(R,m,k)$ is a Noetherian local ring, $M$ is an $R$-finite module whose depth is $n$. One then defines the type of $M$ by the formula (as in the text "Cohen-Macaulay Rings" of Bruns and Herzog):
$$
\tau(M) = \mbox{dim}_k\mbox{Ext}_R^n(k,M).
$$
This definition only makes sense if $\mbox{Ext}_R^n(k,M)$ is a vector space over $k$. One knows that $\mbox{Ext}_R^n(k,M)$ is an $R$-module but why is it a vector space over $k$ then? I guess one needs to verify that $m \subset ann(\mbox{Ext}_R^n(k,M))$ but this is not evident to me at all. Thanks for any comment and answer!