Basis for R-Module R

Theorem

Let $\struct {R, +, \times}$ be a ring with unity whose unity is $1_R$.

Let $\struct {R, +_R, \circ}_R$ denote the $R$-module $R$.

Then $\set {1_R}$ is a basis for $\struct {R, +_R, \circ}_R$.

Proof

From Dimension of $R$-Module $R$ is $1$ we have that $\struct {R, +_R, \circ}_R$ is $1$-dimensional.

From Standard Ordered Basis is Basis it follows directly that $\set {1_R}$ is a basis for $\struct {R, +_R, \circ}_R$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations