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

Definition

Let $R$ be a ring.

Let $\struct {G, +_G, \circ}_R$ and $\struct {H, +_H, \circ}_R$ be $R$-modules.

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

Then $\phi$ is a module isomorphism if and only if $\phi$ is a bijection.

Also see

• Definition:Module Automorphism
• Definition:Module Homomorphism

• Definition:Isomorphism (Abstract Algebra)

• Results about module isomorphisms can be found here.