# 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:

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

This needs considerable tedious hard slog to complete it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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 $40$