Editing Ring theory (section)

===Representation theory===
{{main|Representation theory}}
[[Representation theory]] is a branch of [[mathematics]] that draws heavily on non-commutative rings. It studies [[abstract algebra|abstract]] [[algebraic structure]]s by ''representing'' their [[element (set theory)|elements]] as [[linear transformation]]s of [[vector space]]s, and studies
[[Module (mathematics)|modules]] over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by [[matrix (mathematics)|matrices]] and the [[algebraic operation]]s in terms of [[matrix addition]] and [[matrix multiplication]], which is non-commutative. The [[algebra]]ic objects amenable to such a description include [[group (mathematics)|groups]], [[associative algebra]]s and [[Lie algebra]]s. The most prominent of these (and historically the first) is the [[group representation|representation theory of groups]], in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.