# Ring with Unity is Module over Itself

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## 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.