Dimension of R-Module R is 1

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Theorem

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

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

Then the dimension of $\struct {R, +_R, \circ}_R$ is $1$.


Proof

We have by definition that the $R$-module $R$ is the special case of the $R$-module $R^n$ where $n = 1$.

From $R$-module $R^n$ is $n$-Dimensional it follows that $\struct {R, +_R, \circ}_R$ is $1$-dimensional.

$\blacksquare$