Editing Commutative algebra (section)

{{Short description|Branch of algebra that studies commutative rings}}
{{About|a branch of algebra|algebras that are commutative|Commutative algebra (structure)}}
{{More footnotes|date=June 2019}}
{{Ring theory sidebar}}

[[File:Emmy noether postcard 1915.jpg|thumb|A 1915 postcard from one of the pioneers of commutative algebra, [[Emmy Noether]], to E. Fischer, discussing her work in commutative algebra]]

'''Commutative algebra''', first known as [[ideal theory]], is the branch of [[algebra]] that studies [[commutative ring]]s, their [[ideal (ring theory)|ideals]], and [[module (mathematics)|modules]] over such rings. Both [[algebraic geometry]] and [[algebraic number theory]] build on commutative algebra. Prominent examples of commutative rings include [[polynomial ring]]s; rings of [[algebraic integer]]s, including the ordinary [[integer]]s <math>\mathbb{Z}</math>; and [[p-adic number|''p''-adic integer]]s.<ref>Atiyah and Macdonald, 1969, Chapter 1</ref>

Commutative algebra is the main technical tool of [[algebraic geometry]], and many results and concepts of commutative algebra are strongly related with geometrical concepts.

The study of rings that are not necessarily commutative is known as [[noncommutative algebra]]; it includes [[ring theory]], [[representation theory]], and the theory of [[Banach algebra]]s.