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

Definition

Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ and $\struct {T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}_R$ be $R$-algebraic structures.

Let $\phi: S \to T$ be an $R$-algebraic structure homomorphism.

Then $\phi$ is an $R$-algebraic structure isomorphism if and only if $\phi$ is a bijection.

Module Isomorphism

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.

Vector Space Isomorphism

Let $\struct {V, +, \circ }$ and $\struct {W, +', \circ'}$ be $K$-vector spaces.

Then $\phi: V \to W$ is a vector space isomorphism if and only if:

$(1): \quad \phi$ is a bijection
$(2): \quad \forall \mathbf x, \mathbf y \in V: \map \phi {\mathbf x + \mathbf y} = \map \phi {\mathbf x} +' \map \phi {\mathbf y}$
$(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \map \phi {\lambda \mathbf x} = \lambda \map \phi {\mathbf x}$