Noncommutative ringIn mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings. Sometimes the term noncommutative ring is used instead of ring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative.
Idempotent (ring theory)In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = ... = an for any positive integer n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring.
Noetherian moduleIn abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated.
Semi-local ringIn mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals".
Glossary of ring theoryRing theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. For the items in commutative algebra (the theory of commutative rings), see glossary of commutative algebra. For ring-theoretic concepts in the language of modules, see also Glossary of module theory. For specific types of algebras, see also: Glossary of field theory and Glossary of Lie groups and Lie algebras.
Primitive ringIn the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero. A ring R is said to be a left primitive ring if it has a faithful simple left R-module. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M.
Module artinienEn théorie des anneaux, un module artinien (du nom d'Emil Artin) est un module vérifiant la condition de chaîne descendante. On dit qu'un module M vérifie la condition de chaîne descendante si toute suite décroissante de sous-modules de M est stationnaire. Cela équivaut à dire que tout ensemble non vide de sous-modules de M admet un élément minimal (pour la relation d'inclusion). Tout module fini est artinien. En particulier, tout groupe abélien fini est artinien (en tant que Z-module).
Reduced ringIn ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative ring is reduced if and only if its nilradical is zero.
Algèbre d'AzumayaEn mathématiques, la notion d'algèbre d'Azumaya est une généralisation de la notion d'algèbre centrale simple aux R-algèbres dont les scalaires R ne forment pas un corps. Elle a été introduite dans un article de en 1951, dans le cas où R est un anneau local commutatif, puis a été développée par Alexander Grothendieck comme ingrédient de base à une théorie du groupe de Brauer en géométrie algébrique, dans les séminaires Bourbaki à partir de 1964.
Endomorphism ringIn mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity.
Perfect ringIn the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in Bass's book. A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.
Invariant basis numberIn mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the case of fields, the IBN property becomes the statement that finite-dimensional vector spaces have a unique dimension. A ring R has invariant basis number (IBN) if for all positive integers m and n, Rm isomorphic to Rn (as left R-modules) implies that m = n.
Matrix ringIn abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication . The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs. When R is a commutative ring, the matrix ring Mn(R) is an associative algebra over R, and may be called a matrix algebra.
BimoduleIn abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules. If R and S are two rings, then an R-S-bimodule is an abelian group such that: M is a left R-module and a right S-module.
Théorie des anneauxEn mathématiques, la théorie des anneaux porte sur l'étude de structures algébriques qui imitent et étendent les entiers relatifs, appelées anneaux. Cette étude s'intéresse notamment à la classification de ces structures, leurs représentations, et leurs propriétés. Développée à partir de la fin du siècle, notamment sous l'impulsion de David Hilbert et Emmy Noether, la théorie des anneaux s'est trouvée être fondamentale pour le développement des mathématiques au siècle, au travers de la géométrie algébrique et de la théorie des nombres notamment, et continue de jouer un rôle central en mathématiques, mais aussi en cryptographie et en physique.
K-théorie algébriqueEn mathématiques, la K-théorie algébrique est une branche importante de l'algèbre homologique. Son objet est de définir et d'appliquer une suite de foncteurs K de la catégorie des anneaux dans celle des groupes abéliens. Pour des raisons historiques, K et K sont conçus en des termes un peu différents des K pour n ≥ 2. Ces deux K-groupes sont en effet plus accessibles et ont plus d'applications que ceux d'indices supérieurs. La théorie de ces derniers est bien plus profonde et ils sont beaucoup plus difficiles à calculer, ne serait-ce que pour l'anneau des entiers.
Hereditary ringIn mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required only for finitely generated submodules, it is called semihereditary. For a noncommutative ring R, the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring.
Radical de JacobsonEn algèbre, le radical de Jacobson d'un anneau commutatif est l'intersection de ses idéaux maximaux. Cette notion est due à Nathan Jacobson qui le premier en a fait l'étude systématique. Un élément x appartient au radical de Jacobson de l'anneau A si et seulement si 1 + ax est inversible pour tout a de A. Notons J le radical de Jacobson de l'anneau commutatif A et exploitons le fait que (d'après le théorème de Krull) 1 + ax est non inversible si et seulement s'il appartient à un idéal maximal.
Module simpleUn module M sur un anneau A est dit simple ou irréductible si M n'est pas le module nul et il n'existe pas de sous-modules de M en dehors de {0} et M. Les Z-modules simples sont les groupes abéliens simples, c'est-à-dire les groupes cycliques d'ordre premier. Les espaces vectoriels simples (sur un corps non nécessairement commutatif) sont les droites vectorielles. Étant donné un anneau A et I un idéal à gauche non nul de A, I est un A-module simple si et seulement si I est un idéal minimal à gauche.
Matrice (mathématiques)thumb|upright=1.5 En mathématiques, les matrices sont des tableaux d'éléments (nombres, caractères) qui servent à interpréter en termes calculatoires, et donc opérationnels, les résultats théoriques de l'algèbre linéaire et même de l'algèbre bilinéaire. Toutes les disciplines étudiant des phénomènes linéaires utilisent les matrices. Quant aux phénomènes non linéaires, on en donne souvent des approximations linéaires, comme en optique géométrique avec les approximations de Gauss.
