# Quotient Ring by Null Ideal

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## Theorem

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $\struct {\set {0_R}, +, \circ}$ be the null ideal of $\struct {R, +, \circ}$.

Let $\struct {R / \set {0_R}, +, \circ}$ be the quotient ring of $R$ defined by $\set {0_R}$.

Then $\struct {R / \set {0_R}, +, \circ}$ is isomorphic to $\struct {R, +, \circ}$.

## Proof

Consider the additive group $\struct {R, +}$.

From Trivial Quotient Group is Quotient Group:

- $\struct {R, +} / \set {0_R} \cong \struct {R, +}$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 22$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Example $40$