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

## Definition

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

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

Then $\phi$ is an $R$-algebraic structure isomorphism iff $\phi$ is a bijection.

### Module Isomorphism

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.

### Vector Space Isomorphism

Let $\left({V, +, \circ }\right)$ and $\left({W, +', \circ'}\right)$ be $K$-vector spaces.

Then $\phi: V \to W$ is a vector space isomorphism iff:

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