Is a quotient of a torsion module always a torsion module? Is a direct sum of torsion modules torsion?
I think the second one is yes, but I am not sure about the first one. Can anyone help?
2026-04-24 03:11:03.1777000263
Is a quotient of a torsion module always a torsion module? Is a direct sum of torsion modules torsion?
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Suppose $M$ is a torsion module and $A$ is a submodule. Let $f:M\to M/A$ be a surjective homomorphism. Then for every $m'\in M/A$, there exists an $m\in M$ with $f(m)=m'$. If $r\in R$ is such that $rm=0$, then $rf(m)=f(rm)=0$.