# Ideal of Unit is Whole Ring

## Theorem

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

Let $J$ be an ideal of $R$.

If $J$ contains a unit of $R$, then $J = R$.

### Corollary

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

Let $J$ be an ideal of $R$.

If $J$ contains the unity of $R$, then $J = R$.

## Proof

Let $u \in J$, where $u \in U_R$.

Also by definition, we have $u^{-1} \in U_R$.

Let $x \in R$.

 $\ds$  $\ds x \in R$ $\ds$ $\leadsto$ $\ds x \circ u^{-1} \in R$ as $R$ is closed $\ds$ $\leadsto$ $\ds \paren {x \circ u^{-1} } \circ u \in J$ Definition of Ideal of Ring $\ds$ $\leadsto$ $\ds x \in J$ Ring properties: $u \circ u^{-1} = 1_R$

Thus $R \subseteq J$.

As $J \subseteq R$ by definition, it follows that $J = R$.

$\blacksquare$