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The action of an element $r\in R$ on $\mbox{Ext}_R^n(k,M)$ is the map $\mbox{Ext}_R^n(k,M)\to \mbox{Ext}_R^n(k,M)$ which is induced by either the map $k\to k$ given by multiplication by $r$, or by the map $M\to M$ given by multiplication by $r$ (the two induced maps are the same) Now, ff if $r\in\mathfrak m$ then the map $k\to k$ is zero, so...
The action of an element $r\in R$ on $\mbox{Ext}_R^n(k,M)$ is the map $\mbox{Ext}_R^n(k,M)\to \mbox{Ext}_R^n(k,M)$ which is induced by either the map $k\to k$ given by multiplication by $r$, or by the map $M\to M$ given by multiplication by $r$ (the two induced maps are the same) Now, ff $r\in\mathfrak m$ then the map $k\to k$ is zero, so...