Editing Ring (mathematics) (section)

=== Noncommutative rings ===
* For any ring {{mvar|R}} and any natural number {{mvar|n}}, the set of all square {{mvar|n}}-by-{{mvar|n}} [[matrix (mathematics)|matrices]] with entries from {{mvar|R}}, forms a ring with matrix addition and matrix multiplication as operations. For {{math|1=''n'' = 1}}, this matrix ring is isomorphic to {{mvar|R}} itself. For {{math|''n'' > 1}} (and {{mvar|R}} not the zero ring), this matrix ring is noncommutative.
* If {{math|''G''}} is an [[abelian group]], then the [[group homomorphism|endomorphisms]] of {{math|''G''}} form a ring, the [[endomorphism ring]] {{math|End(''G'')}} of&nbsp;{{math|''G''}}. The operations in this ring are addition and composition of endomorphisms. More generally, if {{mvar|V}} is a [[left module]] over a ring {{mvar|R}}, then the set of all {{mvar|R}}-linear maps forms a ring, also called the endomorphism ring and denoted by {{math|End{{sub|''R''}}(''V'')}}.
*The [[endomorphism ring of an elliptic curve]]. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero.
* If {{math|''G''}} is a [[group (mathematics)|group]] and {{mvar|R}} is a ring, the [[group ring]] of {{math|''G''}} over {{mvar|R}} is a [[free module]] over {{mvar|R}} having {{math|''G''}} as basis. Multiplication is defined by the rules that the elements of {{math|''G''}} commute with the elements of {{mvar|R}} and multiply together as they do in the group {{math|''G''}}.
* The [[ring of differential operators]] (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most [[Banach algebra]]s are noncommutative.