Quotient ring

of a ring $R$ by an ideal $I$

The quotient group of the additive group of $R$ by the subgroup $I$, with multiplication

$$(a+I)(b+I)=ab+I.$$

The quotient turns out to be a ring and is denoted by $R/I$. The mapping $\pi\colon R\to R/I$, where $\pi(x)=x+I$, is a surjective ring homomorphism, called the natural homomorphism (see Algebraic system).

The most important example of a quotient ring is the ring of residues modulo $n$ — the quotient ring of the ring of integers $\mathbf Z$ by the ideal $\mathbf Zn$. The elements of $\mathbf Z/\mathbf Zn$ can be assumed to be the numbers $\{0,\ldots,n-1\}$, where the sum and the product are defined as the remainders on diving the usual sum and product by $n$. One can establish a one-to-one order-preserving correspondence between the ideals of $R/I$ and the ideals of $R$ containing $I$. In particular, $R/I$ is simple (cf. Simple ring) if and only if $I$ is a maximal ideal.

Another most important example is the quotient ring $F[x]/F[x]f(x)$, where $F[x]$ is the ring of polynomials over $F$ in one variable $x$ and $f(x)$ is an irreducible polynomial. This quotient ring describes all field extensions of $F$ by roots of the equation $f(x)=0$ (cf. also Extension of a field).