# Ideal of Unit is Whole Ring

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## 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 $J = R$.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$: Theorem $23.2$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 58.2$ Ideals