Let $M$ be a finitely generated module over a noetherian ring $R$ and suppose that $\dim R≤1$. And let be a one to one homomorphism $f:M\to M$. It is true that $\operatorname{Coker}f$ has finite length as $R$-module?
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A finitely generated module over a noetherian ring is of finite length iff its support is contained into the maximal spectrum. (See here, Proposition 1.6.9.)
Now let $\mathfrak p\in\operatorname{Supp}(\operatorname{Coker}f)$. If $\mathfrak p$ is not maximal, then it is minimal since $\dim R\le 1$. But $(\operatorname{Coker}f)_{\mathfrak p}=\operatorname{Coker}f_{\mathfrak p}$, so the question can be viewed over $R_{\mathfrak p}$. But this is an artinian ring, and then $M_{\mathfrak p}$ is an artinian module, so $f_{\mathfrak p}$ is surjective (see here), that is, $\operatorname{Coker}f_{\mathfrak p}=0$, a contradiction.
