Let $R$ be a commutative ring and $M,N$ $R$-modules finitely generated with $M$ of finite length. How can I prove that $M\otimes_R N$ is of finite length?
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Let $F$ be a finite free $R$-module and let $F \to N$ be a surjective map of $R$-modules. Tensoring with $M$, you get a surjective $R$-linear map $M \otimes_R F \to M \otimes_R N $. Notice that $M \otimes_R F \simeq M \oplus \cdots \oplus M$ has finite length.