# Quotient Ring is Ring/Quotient Ring Addition is Well-Defined

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## Theorem

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$.

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

Let $\struct {R / J, +, \circ}$ be the quotient ring of $R$ by $J$.

Then $+$ is well-defined on $R / J$, that is:

- $x_1 + J = x_2 + J, y_1 + J = y_2 + J \implies \paren {x_1 + y_1} + J = \paren {x_2 + y_2} + J$

## Proof

From Ideal is Additive Normal Subgroup that $J$ is a normal subgroup of $R$ under $+$.

Thus, the quotient group $\struct {R / J, +}$ is defined, and as a Quotient Group is Group, $+$ is well-defined.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Theorem $41$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms