# Quotient rings

We can also define the quotient ring (AKA factor ring) of a ring $${R}$$, and related concepts:

• Ideal: additive subgroup $${A\subseteq R}$$ where $${ra,ar\in A\;\forall a\in A,\, r\in R}$$
• Quotient ring: the cosets $${R/A\equiv\left\{ r+A\mid r\in R\right\} }$$, which form a ring iff $${A}$$ is an ideal
• Prime ideal: proper ideal $${A\subset R\mid ab\in A}$$ for $${a,b\in R\Rightarrow a\in A}$$ or $${b\in A}$$
• Maximal ideal: $${\forall}$$ ideal $${B\supseteq A}$$, $${B=A}$$ or $${B=R}$$

The definition of ideal above is sometimes called a two-sided ideal, in which case a left ideal only requires that $${ra\in A}$$ and a right ideal requires that $${ar\in A}$$. For a commutative ring, these are all equivalent. These concepts are also applied to associative algebras, since with scalars ignored they are rings.

Note that since a ring is an abelian group under addition, every subgroup is already normal. As with groups, the kernel of a ring homomorphism $${\phi}$$ is an ideal, and factors $${R}$$ into elements isomorphic to those of the image of $${R}$$: $${R/\textrm{Ker}\phi\cong\phi(R)}$$. Some additional related facts are:

• For $${R}$$ commutative with unity, $${R/A}$$ is an integral domain iff $${A}$$ is prime
• For $${R}$$ commutative with unity, $${R/A}$$ is a field if $${A}$$ is maximal

Continuing to add structure, in a vector space $${V}$$ we can take the quotient $${V/W}$$ for any subspace $${W}$$, which is just isomorphic to the orthogonal complement of $${W}$$ in $${V}$$.