For questions related to module isomorphism. Let $R$ be a ring, $(G,+G,∘)_R$ and $(H,+H,∘)_R$ be $R$-modules and $ϕ:G→H$ be a module homomorphism. Then $ϕ$ is a module isomorphism iff $ϕ$ is a bijection.
A module homomorphism is called a module isomorphism if it admits an inverse homomorphism; in particular, it is a bijection. Conversely, one can show a bijective module homomorphism is an isomorphism; i.e., the inverse is a module homomorphism. In particular, a module homomorphism is an isomorphism if and only if it is an isomorphism between the underlying abelian groups.
Let $R$ be a ring. Let $(G,+G,∘)_R$ and $(H,+H,∘)_R$ be $R$-modules. Let $ϕ:G→H$ be a module homomorphism.
Then $ϕ$ is a module isomorphism iff $ϕ$ is a bijection.
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