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Ring theory - Wikipedia
In algebra, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings (group … See more
A ring is called commutative if its multiplication is commutative. Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to formalize properties of the See more
Noncommutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining noncommutative geometry based on noncommutative rings. Noncommutative rings and See more
The ring of integers of a number field
The coordinate ring of an algebraic variety
If X is an affine algebraic variety, then the set of all regular … See moreearly 19th centuryCommutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory.1920Emmy Noether, in collaboration with W. Schmeidler, published a paper about the theory of ideals in which they defined left and right ideals in a ring.1921Emmy Noether published a landmark paper called Idealtheorie in Ringbereichen, analyzing ascending chain conditions with regard to (mathematical) ideals.1928Emil Artin generalized Wedderburn's structure theorems to Artinian rings.1980sThe trend of building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (non-existent) 'noncommutative spaces' started.1989K. R. Goodearl and R. B. Warfield Jr. published An Introduction to Noncommutative Noetherian Rings.1991R. B. J. T. Allenby published Rings, Fields and Groups.1999Carl Faith published Rings and Things and a Fine Array of Twentieth Century Associative Algebra.1999Hideyuki Matsumura published Commutative Ring Theory.2001T. Y. Lam published A First Course in Noncommutative Rings.General
• Isomorphism theorems for rings
• Nakayama's lemma
Structure theorems
• The See moreDimension of a commutative ring
In this section, R denotes a commutative ring. The Krull dimension of R is the supremum of the lengths n of all the chains of prime ideals See moreCommutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. Central to the development of these subjects were the rings of integers … See more
Wikipedia text under CC-BY-SA license 16.1: Rings, Basic Definitions and Concepts - Mathematics …
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