Ring theory
Noncommutative rings 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 functions on (non-existent) 'noncommutative spaces'.This trend started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups.Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility.Alexander Grothendieck completed this by introducing schemes, a generalization of algebraic varieties, which may be built from any commutative ring.These objects are the "affine schemes" (generalization of affine varieties), and a general scheme is then obtained by "gluing together" (by purely algebraic methods) several such affine schemes, in analogy to the way of constructing a manifold by gluing together the charts of an atlas.Representation theory is a branch of mathematics that draws heavily on non-commutative rings.It studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication, which is non-commutative.General Structure theorems Other 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 idealsRatliff proved that a noetherian local integral domain R is catenary if and only if for every prime ideal[3] If R is an integral domain that is a finitely generated k-algebra, then its dimension is the transcendence degree of its field of fractions over k. If S is an integral extension of a commutative ring R, then S and R have the same dimension.Closely related concepts are those of depth and global dimension.Two rings R, S are said to be Morita equivalent if the category of left modules over R is equivalent to the category of left modules over S. In fact, two commutative rings which are Morita equivalent must be isomorphic, so the notion does not add anything new to the category of commutative rings.Morita equivalence is especially important in algebraic topology and functional analysis.the set of isomorphism classes of finitely generated projective modules over R; let alsoIn algebraic number theory, R will be taken to be the ring of integers, which is Dedekind and thus regular.The fact that the Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by its modules.Noncommutative rings are an active area of research due to their ubiquity in mathematics.For instance, the ring of n-by-n matrices over a field is noncommutative despite its natural occurrence in geometry, physics and many parts of mathematics.This may be seen via either Hilbert's Nullstellensatz or scheme-theoretic constructions (i.e., Spec and Proj).The fundamental theorem of symmetric polynomials states that this ring isCentral 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.More precisely, William Rowan Hamilton put forth the quaternions and biquaternions; James Cockle presented tessarines and coquaternions; and William Kingdon Clifford was an enthusiast of split-biquaternions, which he called algebraic motors.One sign of re-organization was the use of direct sums to describe algebraic structure.The various hypercomplex numbers were identified with matrix rings by Joseph Wedderburn (1908) and Emil Artin (1928).Wedderburn's structure theorems were formulated for finite-dimensional algebras over a field while Artin generalized them to Artinian rings.The following year she published a landmark paper called Idealtheorie in Ringbereichen, analyzing ascending chain conditions with regard to (mathematical) ideals.Noted algebraist Irving Kaplansky called this work "revolutionary";[8] the publication gave rise to the term "Noetherian ring", and several other mathematical objects being called Noetherian.