Definition:Module on Cartesian Product

Definition

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $n \in \N_{>0}$.

Let $+: R^n \times R^n \to R^n$ be defined as:

$\tuple {\alpha_1, \ldots, \alpha_n} + \tuple {\beta_1, \ldots, \beta_n} = \tuple {\alpha_1 +_R \beta_1, \ldots, \alpha_n +_R \beta_n}$

Let $\times: R \times R^n \to R^n$ be defined as:

$\lambda \times \tuple {\alpha_1, \ldots, \alpha_n} = \tuple {\lambda \times_R \alpha_1, \ldots, \lambda \times_R \alpha_n}$

Then $\struct {R^n, +, \times}_R$ is the $R$-module $R^n$.

Special Case

A special case of this is for when $n = 1$:

Let $+: R \times R \to R$ be defined as:

$\alpha + \beta = \alpha +_R \beta$

Let $\times: R \times R \to R$ be defined as:

$\lambda \times \alpha = \lambda \times_R \alpha$

Then $\struct {R, +, \times}_R$ is the $R$-module $R$.

Also see

• Module on Cartesian Product is Module

• Results about Module on Cartesian Product can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.1$