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In mathematics, an associative algebraA is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field K. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.
In mathematics, more specifically ring theory, the Jacobson radical of a ring is the ideal consisting of those elements in that annihilate all simple right -modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by or ; the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in.
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.
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, 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.
Ring 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.
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type.
In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i. In particular, any simple left or right Artinian ring is isomorphic to an n-by-n matrix ring over a division ring D, where both n and D are uniquely determined.
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
In mathematics, specifically abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.
In 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. (Lam 2001, p. §20)(Mikhalev & Pilz 2002, p. C.7)
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings.
In mathematics, the term socle has several related meanings.
In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical. If the algebra is finite-dimensional this is equivalent to saying that it can be expressed as a Cartesian product of simple subalgebras.
In the mathematical field of Lie theory, the radical of a Lie algebra is the largest solvable ideal of
In 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.
In 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.
In abstract algebra, a uniserial moduleM is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either or . A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in an analogous way, and are in general distinct from their right counterparts.
In mathematics, especially ring theory, the class of Frobenius rings and their generalizations are the extension of work done on Frobenius algebras. Perhaps the most important generalization is that of quasi-Frobenius rings, which are in turn generalized by right pseudo-Frobenius rings and right finitely pseudo-Frobenius rings. Other diverse generalizations of quasi-Frobenius rings include QF-1, QF-2 and QF-3 rings.
This is a glossary of commutative algebra.
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context.
References
- David Eisenbud, Commutative algebra with a view toward Algebraic Geometry ISBN 0-387-94269-6
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