# Ring with Unity is Module over Itself

## Theorem

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

Then $\struct {R, +, \circ}_R$ is a unitary $R$-module.

## Proof

From Ring is Module over Itself we have that $\struct {R, +, \circ}_R$ is an $R$-module.

We have by hypothesis that $\struct {R, +, \circ}$ has a unity $1_R$.

For $\struct {R, +, \circ}_R$ to be unitary, it must satisfy the additional axiom:

 $(4)$ $:$ $\ds \forall x \in R:$ $\ds 1_R \circ x = x$

The axiom follows from the definition of a unity.