Editing Ring (mathematics) (section)

== Rings with extra structure ==
A ring may be viewed as an [[abelian group]] (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
* An [[associative algebra]] is a ring that is also a [[vector space]] over a field {{mvar|n}} such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of {{mvar|n}}-by-{{mvar|n}} matrices over the real field {{tmath|\R}} has dimension {{math|''n''{{sup|2}}}} as a real vector space.
* A ring {{mvar|R}} is a [[topological ring]] if its set of elements {{mvar|R}} is given a [[topological space|topology]] which makes the addition map (<math>+ : R\times R \to R</math>) and the multiplication map {{math|⋅ : ''R'' × ''R'' → ''R''}} to be both [[Continuous function (topology)|continuous]] as maps between topological spaces (where {{math|''X'' × ''X''}} inherits the [[product topology]] or any other product in the category). For example, {{mvar|n}}-by-{{mvar|n}} matrices over the real numbers could be given either the [[Euclidean topology]], or the [[Zariski topology]], and in either case one would obtain a topological ring.
* A [[λ-ring]] is a commutative ring {{mvar|R}} together with operations {{math|''λ''{{sup|''n''}}: ''R'' → ''R''}} that are like {{mvar|n}}th [[exterior power]]s:
*: <math>\lambda^n(x + y) = \sum_0^n \lambda^i(x) \lambda^{n-i}(y).</math>
: For example, {{tmath|\Z}} is a λ-ring with <math>\lambda^n(x) = \binom{x}{n},</math> the [[binomial coefficient]]s. The notion plays a central rule in the algebraic approach to the [[Riemann–Roch theorem]].
* A [[totally ordered ring]] is a ring with a [[total ordering]] that is compatible with ring operations.<!-- Z is characterized as a certain unique ordered ring. Can't remember the precise statement.-->