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 \({V}\).

An Illustrated Handbook