Ring is Ideal of Itself

Theorem

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

Then $R$ is an ideal of $R$.

Proof

From Null Ring and Ring Itself Subrings, $\struct {R, +, \circ}$ is a subring of $\struct {R, +, \circ}$.

Also:

$\forall x, y \in \struct {R, +, \circ}: x \circ y \in R$

thus fulfilling the condition for $R$ to be an ideal of $R$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Example $40$
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms: Definition $2.5$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 58.1$ Ideals