Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Module Isomorphism

Definition

Let $R$ be a ring.

Let $\left({G, +_G, \circ}\right)_R$ and $\left({H, +_H, \circ}\right)_R$ be $R$-modules.

Let $\phi: G \to H$ be a module homomorphism.

Then $\phi$ is a module isomorphism iff $\phi$ is a bijection.