> [!NOTE] **Definition** (Quotient Ring) > Let $(R,+,\times)$ be [[Rings|ring]]. Let $J$ be an [[Ideal of Ring|ideal]] of $R$. Let $R/J$ be the [[Coset space|coset space]] of $R$ modulo $J$ with respect to $+.$ We define addition on $R/J$ by $(x+I)+(y+I)=(x+y)+I$and multiplication on $R/J$ by $(x+I)(y+I)=(x \times y)+I$The [[Algebraic Structure|algebraic structure]] $(R/J,+,\times)$ is the quotient ring of $R$ by $J.$ > [!Example] Examples > Let $n\in \mathbb{Z}.$ The [[Integers modulo n|integers modulo n]] is the quotient ring of $\mathbb{Z}$ by $n\mathbb{Z}$. # Properties Note that [[Quotient Ring Operations are Well-Defined|quotient ring operations are well-defined]]: that is if $x_{1}-x_{2},y_{1}-y_{2}\in I$ then $(x_{1}+I)+(y_{1}+I)=(x_{2}+I)+(y_{2}+I)$ and $(x_{1}+I)(y_{1}+I)=(x_{2}+I)(y_{2}+I).$ Thus $(R/ I,+,\times)$ is [[Quotient Ring is Ring|indeed a ring]].