Editing Ring theory (section)

==History==
Commutative ring theory originated in algebraic number theory, algebraic geometry, and [[invariant theory]]. Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables. Noncommutative ring theory began with attempts to extend the complex numbers to various [[hypercomplex number]] systems. The genesis of the theories of commutative and noncommutative rings dates back to the early 19th century, while their maturity was achieved only in the third decade of the 20th century.

More precisely, [[William Rowan Hamilton]] put forth the [[quaternion]]s and [[biquaternion]]s; [[James Cockle (lawyer)|James Cockle]] presented [[tessarine]]s and [[coquaternion]]s; and [[William Kingdon Clifford]] was an enthusiast of [[split-biquaternion]]s, which he called ''algebraic motors''. These noncommutative algebras, and the non-associative [[Lie algebra]]s, were studied within [[universal algebra]] before the subject was divided into particular [[mathematical structure]] types. One sign of re-organization was the use of [[direct sum of modules#Direct sum of algebras|direct sums]] to describe algebraic structure.

The various hypercomplex numbers were identified with [[matrix ring]]s by [[Joseph Wedderburn]] (1908) and [[Emil Artin]] (1928). Wedderburn's structure theorems were formulated for finite-dimensional [[algebra over a field|algebras over a field]] while Artin generalized them to [[Artinian ring]]s.

In 1920, [[Emmy Noether]], in collaboration with W. Schmeidler, published a paper about the [[ideal theory|theory of ideals]] in which they defined [[Ideal (ring theory)|left and right ideals]] in a [[ring (mathematics)|ring]]. The following year she published a landmark paper called ''Idealtheorie in Ringbereichen'', analyzing [[ascending chain condition]]s with regard to (mathematical) ideals. Noted algebraist [[Irving Kaplansky]] called this work "revolutionary";{{Sfn |Kimberling|1981|p=18}} the publication gave rise to the term "[[Noetherian ring]]", and several other mathematical objects being called ''[[Noetherian (disambiguation)|Noetherian]]''.{{Sfn |Kimberling|1981|p=18}}<ref>{{citation|last= Dick|first= Auguste|author-link=Auguste Dick|title= Emmy Noether: 1882–1935| publisher= [[Birkhäuser]] | year = 1981| isbn =3-7643-3019-8 | translator-first= H. I. | translator-last= Blocher}}, p. 44–45.</ref>