Ideal of Unit is Whole Ring/Corollary

Corollary to Ideal of Unit is Whole Ring

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

Follows directly from Ideal of Unit is Whole Ring and Unity is Unit.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Theorem $35$
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2$: Exercise $1$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 58.2$ Ideals