Quotient ring

Definition

Let $R$ be a ring and let $I$ be a two-sided ideal of $R$. For $a \in R$, the coset of $a$ is $a + I = \{a + x : x \in I\}$. The quotient ring $R/I$ is the set of the cosets of $I$, with these operations: $$(a + I) + (b + I) = (a + b) + I, \qquad (a + I)(b + I) = a b + I.$$ These operations do not depend on the choice of $a$ and $b$ in their cosets. With them, $R/I$ is a ring with zero $0 + I = I$ and identity $1 + I$.

Remarks

Two elements $a, b \in R$ have the same coset if and only if $a - b \in I$. The map $\pi \colon R \to R/I$, $\pi(a) = a + I$, is a surjective ring homomorphism with kernel $I$. For $n \ge 1$, the quotient $\mathbb{Z}/n\mathbb{Z}$ is the ring of integers modulo $n$. The quotient $R/R$ is the zero ring.

Explanations

L1 human

To make the quotient ring, you declare all the elements of $I$ to be zero. Two elements become equal when their difference is in $I$. For $\mathbb{Z}/5\mathbb{Z}$, the numbers $2$ and $7$ become the same element.