Algebras, Rings and Modules: Non-commutative Algebras and Rings

Noncommutative geometry
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There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a Serre's theorem on Proj. Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length; there is also analogous theorem for coherent sheaves when the algebra is Noetherian. This theorem is extended as a definition of noncommutative projective geometry by Michael Artin and J.

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Zhang, [3] who add also some general ring-theoretic conditions e. Artin-Schelter regularity. Many properties of projective schemes extend to this context. For example, there exist an analog of the celebrated Serre duality for noncommutative projective schemes of Artin and Zhang. Rosenberg has created a rather general relative concept of noncommutative quasicompact scheme over a base category , abstracting the Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors.

Some of the motivating questions of the theory are concerned with extending known topological invariants to formal duals of noncommutative operator algebras and other replacements and candidates for noncommutative spaces. One of the main starting points of the Alain Connes ' direction in noncommutative geometry is his discovery of a new homology theory associated to noncommutative associative algebras and noncommutative operator algebras, namely the cyclic homology and its relations to the algebraic K-theory primarily via Connes-Chern character map.

The theory of characteristic classes of smooth manifolds has been extended to spectral triples, employing the tools of operator K-theory and cyclic cohomology.

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Several generalizations of now classical index theorems allow for effective extraction of numerical invariants from spectral triples. The fundamental characteristic class in cyclic cohomology, the JLO cocycle , generalizes the classical Chern character.

From Wikipedia, the free encyclopedia. Douglas, Albert Schwarz, Noncommutative geometry and matrix theory: compactification on tori. High Energy Phys. Artin, J. Zhang, Noncommutative projective schemes, Adv. Zhang, Serre duality for noncommutative projective schemes, Proc. Willaert, Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras, J.

Pure Appl. Snyder, Quantized Space-Time, Phys. Categories : Noncommutative geometry Mathematical quantization Quantum gravity. Namespaces Article Talk. Views Read Edit View history. In other projects Wikiquote. By using this site, you agree to the Terms of Use and Privacy Policy. Algebraic number field. 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.

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Ring theory studies the structure of rings, their representations , or, in different language, modules , special classes of rings group rings , division rings , universal enveloping algebras , as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory , which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra , a major area of modern mathematics. Because these three fields algebraic geometry, algebraic number theory and commutative algebra are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to.

[] The non-commutative scheme having a free algebra as a homogeneous coordinate ring

For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly, Fermat's last theorem is stated in terms of elementary arithmetic , which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.

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 s with the development of noncommutative geometry and with the discovery of quantum groups. It has led to a better understanding of noncommutative rings, especially noncommutative Noetherian rings. For the definitions of a ring and basic concepts and their properties, see ring mathematics. The definitions of terms used throughout ring theory may be found in the glossary of ring theory.

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 integers. Commutative rings are also important in algebraic geometry. In commutative ring theory, numbers are often replaced by ideals , and the definition of the prime ideal tries to capture the essence of prime numbers. 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.

Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers.

Ring Definition (expanded) - Abstract Algebra

Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. Important examples of commutative rings can be constructed as rings of polynomials and their factor rings. Algebraic geometry is in many ways the mirror image of commutative algebra. This correspondence started with Hilbert's Nullstellensatz that establishes a one-to-one correspondence between the points of an algebraic variety , and the maximal ideals of its coordinate ring.

This correspondence has been enlarged and systematized for translating and proving most geometrical properties of algebraic varieties into algebraic properties of associated commutative rings. Alexander Grothendieck completed this by introducing schemes , a generalization of algebraic varieties, which may be built from any commutative ring. More precisely, the spectrum of a commutative ring is the space of its prime ideals equipped with Zariski topology , and augmented with a sheaf of rings.

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. 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 associative algebras rings that are also vector spaces are often studied via their categories of modules. A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms , very much akin to the way fields integral domains in which every non-zero element is invertible act on vector spaces. Examples of noncommutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings. 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.

The algebraic objects amenable to such a description include groups , associative algebras and Lie algebras. The most prominent of these and historically the first is the representation theory of groups , in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.

Practically all noetherian rings that appear in application are catenary. It is a deep theorem of Ratliff that the converse is also true. 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.

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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. In general, if R is a noetherian local ring, then the depth of R is less than or equal to the dimension of R. When the equality holds, R is called a Cohen—Macaulay ring. A regular local ring is an example of a Cohen—Macaulay ring. It is a theorem of Serre that R is a regular local ring if and only if it has finite global dimension and in that case the global dimension is the Krull dimension of R.

The significance of this is that a global dimension is a homological notion. 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. However, commutative rings can be Morita equivalent to noncommutative rings, so Morita equivalence is coarser than isomorphism. Morita equivalence is especially important in algebraic topology and functional analysis.

It is an abelian group called the Picard group of R. If R is a regular domain i. For example, if R is a principal ideal domain, then Pic R vanishes. In algebraic number theory, R will be taken to be the ring of integers , which is Dedekind and thus regular. It follows that Pic R is a finite group finiteness of class number that measures the deviation of the ring of integers from being a PID.

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In fact, we can see that j is up to scaling the only normal element of degree 3. The structure of a noncommutative ring is more complicated than that of a commutative ring. Thus we conclude M is the third syzygy module and N is the fourth. Viewed 4k times. The various hypercomplex numbers were identified with matrix rings by Joseph Wedderburn and Emil Artin