Dimension of R-Module R is 1

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

Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: This proof applies only for a ring with unity. Needs to be expanded to any ring. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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.

$\blacksquare$