Editing Ring theory (section)

{{about|a mathematical concept|}}
{{short description|Branch of algebra}}
{{Algebraic structures|ring}}
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In [[algebra]], '''ring theory''' is the study of [[ring (mathematics)|rings]], [[algebraic structure]]s in which addition and multiplication are defined and have similar properties to those operations defined for the [[integer]]s. Ring theory studies the structure of rings; their [[representation of an algebra|representations]], or, in different language, [[module (ring theory)|modules]]; special classes of rings ([[group ring]]s, [[division ring]]s, [[universal enveloping algebra]]s); related structures like [[rng (algebra)|rngs]]; as well as an array of properties that prove to be of interest both within the theory itself and for its applications, such as [[homological algebra|homological properties]] and [[Polynomial identity ring|polynomial identities]].

[[Commutative ring]]s are much better understood than noncommutative ones. [[Algebraic geometry]] and [[algebraic number theory]], which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of ''[[commutative algebra]]'', a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. For example, [[Hilbert's Nullstellensatz]] is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly, [[Fermat's Last Theorem]] is stated in terms of elementary [[arithmetic]], which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.

[[Noncommutative ring]]s are quite different in flavour, since more unusual behavior can arise. While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of [[function (mathematics)|function]]s on (non-existent) 'noncommutative spaces'. This trend started in the 1980s with the development of [[noncommutative geometry]] and with the discovery of [[quantum group]]s. It has led to a better understanding of noncommutative rings, especially noncommutative [[Noetherian ring]]s.{{sfnp|Goodearl| Warfield|1989}}

For the definitions of a ring and basic concepts and their properties, see ''[[Ring (mathematics)]]''. The definitions of terms used throughout ring theory may be found in ''[[Glossary of ring theory]]''.