Quick question:
With $R=\mathbb{Z}, M=\mathbb{Z}, P=2\mathbb{Z}, Q=3\mathbb{Z}$ modules, can I conclude that $M/P = \mathbb{Z}_2 \ncong M/Q = \mathbb{Z}_3$ because they have a different number of elements? Since modules are abelian groups?
2026-05-05 08:32:50.1777969970
Two modules not isomorphic if different number of elements?
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A module homomorphism is by definition an isomorphism if it has an inverse that is a homomorphism. On the level of sets, any function with an inverse is a bijection, so isomorphic modules must have the same cardinality.
Note that this should hold for any reasonable algebraic object.