Let $R$ be a ring and $I$ an ideal in $R$. Define an equivalence relation $\sim$ on $R$ by $a\sim b$ when $a-b \in I$. The equivalence classes are given by $[a] = a+ I = {a+r \mid r\in I}$. The set of all such equivalence classes is denoted $R/I$, or the quotient of $R$ by $I$. The operations addition ($(a+I)+ (b+I) = (a+b)+I$) and multiplication ($(a+I)(b+I) = (ab)+I$) are well-defined. write_proof
Moreover, this itself is a ring write_proof.
The quotient comes with a surjective homomorphism $R\to R/I$ given by $r\mapsto r+I$.
Wikidata ID: Q619436