"Two ways" to reduce a module

Question

Let $M$ be a module over a principal ideal domain $R$ and $\mathfrak{m}$ a maximal ideal of $R$ with residue field $R/\mathfrak{m}=k$ of characteristic $p$.

Under what circumstances are the modules $$ M\otimes_R k\quad\mbox{and}\quad M/\mathfrak{m}M $$ isomorphic?

Answer 1

Let $M$ be a module over a commutative ring $R$. Let $I$ be an ideal of $R$.

The following sequence of $R$-modules is exact.

$$0 \rightarrow I \rightarrow R \rightarrow R/I \rightarrow 0$$

Since the functor $M\otimes_R -$ is right exact, the following sequence of $R$-modules is exact.

$$M\otimes_R I \rightarrow M \rightarrow M\otimes_R (R/I) \rightarrow 0$$

Hence $M\otimes_R (R/I)$ is isomorphic to $M/IM$.