Ideal of Unit is Whole Ring

Theorem

Let $\left({R, +, \circ}\right)$ be a ring with unity.

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

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

Corollary

Let $\left({R, +, \circ}\right)$ 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 $x \in R$.

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle x \in R\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle x \circ u^{-1} \in R\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	as $R$ is closed
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left({x \circ u^{-1} }\right) \circ u \in J\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of an Ideal
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle x \in J\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Ring properties: $u \circ u^{-1} = 1_R$

Thus $R \subseteq J$.

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

$\blacksquare$

Proof of Corollary

Follows directly from the main result and the fact that Unity is a Unit.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 23$: Theorem $23.2$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 5.21$: Theorem $35$ (for corollary)
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 2$: Exercise $1$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 58.2$