Editing Ring (mathematics) (section)

== Some examples of the ubiquity of rings ==
Many different kinds of [[mathematical object]]s can be fruitfully analyzed in terms of some [[functor|associated ring]].

=== Cohomology ring of a topological space ===
To any [[topological space]] {{mvar|X}} one can associate its integral [[cohomology ring]]
:<math>H^*(X,\Z ) = \bigoplus_{i=0}^{\infty} H^i(X,\Z ),</math>
a [[graded ring]]. There are also [[homology group]]s <math>H_i(X,\Z )</math> of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the [[sphere]]s and [[torus|tori]], for which the methods of [[point-set topology]] are not well-suited. [[Cohomology group]]s were later defined in terms of homology groups in a way which is roughly analogous to the dual of a [[vector space]]. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the [[universal coefficient theorem]]. However, the advantage of the cohomology groups is that there is a [[cup product|natural product]], which is analogous to the observation that one can multiply pointwise a {{mvar|k}}-[[multilinear form]] and an {{mvar|l}}-multilinear form to get a ({{math|''k'' + ''l''}})-multilinear form.

The ring structure in cohomology provides the foundation for [[characteristic class]]es of [[fiber bundle]]s, intersection theory on manifolds and [[algebraic variety|algebraic varieties]], [[Schubert calculus]] and much more.

=== Burnside ring of a group ===
To any [[group (mathematics)|group]] is associated its [[Burnside ring]] which uses a ring to describe the various ways the group can [[Group action (mathematics)|act]] on a finite set. The Burnside ring's additive group is the [[free abelian group]] whose basis is the set of transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the [[representation ring]]: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.

=== Representation ring of a group ring ===
To any [[group ring]] or [[Hopf algebra]] is associated its [[representation ring]] or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from [[character theory]], which is more or less the [[Grothendieck group]] given a ring structure.

=== Function field of an irreducible algebraic variety ===
To any irreducible [[algebraic variety]] is associated its [[function field of an algebraic variety|function field]]. The points of an algebraic variety correspond to [[valuation ring]]s contained in the function field and containing the [[coordinate ring]]. The study of [[algebraic geometry]] makes heavy use of [[commutative algebra]] to study geometric concepts in terms of ring-theoretic properties. [[Birational geometry]] studies maps between the subrings of the function field.

=== Face ring of a simplicial complex ===
Every [[simplicial complex]] has an associated face ring, also called its [[Stanley–Reisner ring]]. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in [[algebraic combinatorics]]. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of [[simplicial polytope]]s.