Suppose $M$ is a module over some ring $R$ such that $M\cong M_1\oplus M_2$. If $M,\,M_2$ are both torsion-free modules, does it follow that $M_1$ is also? Intuitively I would expect this to be the case, for if $M_1$ had a torsion element, then surely that means $M$ would also?
2026-03-31 15:38:36.1774971516
If $M\cong M_1\oplus M_2$ is a torsion-free module, does it follow that $M_1$ is a torsion-free module?
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You are right. If $m\in M_1$ were a torsion element, then $(m, 0)\in M$ would be one too.