Ring theory
Study of algebraic structures with addition and multiplication.
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. It examines the structure of rings, their representations (modules), special classes of rings (group rings, division rings, universal enveloping algebras), related structures like rngs, and properties such as homological properties and polynomial identities.
- field
- Algebra
- known_for
- Study of rings, modules, commutative and noncommutative rings, algebraic geometry, representation theory
Lore & Background
Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory have driven much of the development of commutative ring theory, now called commutative algebra. Hilbert's Nullstellensatz is fundamental for algebraic geometry and is stated and proved in terms of commutative algebra. Noncommutative rings are quite different, with more unusual behavior. A trend since the 1980s has sought to parallel commutative development by building the theory of certain noncommutative rings in a geometric fashion, as if they were rings of functions on 'noncommutative spaces', leading to better understanding of noncommutative Noetherian rings.
Reader's Guide
Ring theory is a central area of modern mathematics, with commutative rings forming the foundation of commutative algebra, algebraic geometry, and algebraic number theory. The correspondence between algebraic varieties and commutative rings, systematized through schemes, allows geometric properties to be translated into algebraic ones. Noncommutative rings, resembling rings of matrices, are studied via their categories of modules and have inspired noncommutative geometry. Representation theory draws heavily on noncommutative rings, representing abstract structures by matrices. Key theorems include the Artin–Wedderburn theorem for semisimple rings, the Jacobson density theorem for primitive rings, and the Skolem–Noether theorem for automorphisms of simple rings. The Krull dimension of a commutative ring is defined by chains of prime ideals, and the fundamental theorem of dimension theory relates it to generators of primary ideals and the graded ring.
Did You Know?
- Commutative rings are much better understood than noncommutative ones.
- Hilbert's Nullstellensatz establishes a one-to-one correspondence between points of an algebraic variety and maximal ideals of its coordinate ring.
- Noncommutative rings resemble rings of matrices in many respects.
- Wedderburn's little theorem states that finite domains are fields.
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