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Top (algebra)
In the context of a module ''M'' over a ring ''R'', the top of ''M'' is the largest semisimple quotient module of ''M'' if it exists. For finite-dimensional ''k''-algebras (''k'' a field) R, if rad(''M'') denotes the intersection of all proper maximal submodules of ''M'' (the radical of the module), then the top of ''M'' is ''M''/rad(''M''). In the case of local rings with maximal ideal ''P'', the top of ''M'' is ''M''/''PM''. In general if ''R'' is a semilocal ring (=semi-artinian ring), that is, if ''R''/Rad(''R'') is an Artinian ring, where Rad(''R'') is the Jacobson radical of ''R'', then ''M''/rad(''M'') is a semisimple module and is the top of ''M''. This includes the cases of local rings and finite dimensional algebras over fields. See also * Projective cover *Radical of a module *Socle (mathematics) In mathematics, the term socle has several related meanings. Socle of a group In the context of group theory, the socle of a group ''G'', denoted soc(''G''), is the subgro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Module
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. For a group-theory analog of the same notion, see ''Semisimple representation''. Definition A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules. For a module ''M'', the following are equivalent: # ''M'' is semisimple; i.e., a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Module
In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by the fact that in these cases, the subspace that is used for defining the quotient is not of the same nature as the ambient space (that is, a quotient ring is the quotient of a ring by an ideal, not a subring, and a quotient group is the quotient of a group by a normal subgroup, not by a general subgroup). Given a module over a ring , and a submodule of , the quotient space is defined by the equivalence relation : a \sim b if and only if b - a \in B, for any in . The elements of are the equivalence classes = a+B = \. The function \pi: A \to A/B sending in to its equivalence class is called the ''quotient map'' or the ''projection map'', and is a module homomorphism. The addition operation on is defined for two equivalence clas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Submodule
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals contained between ''I'' and ''R''. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal ''A'' is not necessarily two-sided, the quotient ''R''/''A'' is not necessarily a ring, but it is a simple module over ''R''. If ''R'' has a unique maximal right ideal, then ''R'' is known as a local ring, and the maximal right ideal is also the unique maximal lef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of A Module
In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(''M'') of ''M''. Definition Let ''R'' be a ring and ''M'' a left ''R''-module. A submodule ''N'' of ''M'' is called maximal or cosimple if the quotient ''M''/''N'' is a simple module. The radical of the module ''M'' is the intersection of all maximal submodules of ''M'', :\mathrm(M) = \bigcap_ N Equivalently, :\mathrm(M) = \sum_ S These definitions have direct dual analogues for soc(''M''). Properties * In addition to the fact rad(''M'') is the sum of superfluous submodules, in a Noetherian module rad(''M'') itself is a superfluous submodule. * A ring for which rad(''M'') = for every right ''R''-module ''M'' is called a right V-ring. * For any module ''M'', rad(''M''/rad(''M'')) is zero. * ''M'' is a finitely generate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semilocal Ring
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''. 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". Some literature refers to a commutative semi-local ring in general as a ''quasi-semi-local ring'', using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals. A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal. Examples * Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local. * The quotient \mathbb/m\mathbb is a semi-local ring. In particular, if m is a prime power, then \mathbb/m\mathbb is a local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin 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. Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Artin–Wedderburn theorem charact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobson Radical
In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-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 J(R) or \operatorname(R); 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 . The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring and module theoretic results, such as Nakayama's lemma. Definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Cover
In the branch of abstract mathematics called category theory, a projective cover of an object ''X'' is in a sense the best approximation of ''X'' by a projective object ''P''. Projective covers are the dual of injective envelopes. Definition Let \mathcal be a category and ''X'' an object in \mathcal. A projective cover is a pair (''P'',''p''), with ''P'' a projective object in \mathcal and ''p'' a superfluous epimorphism in Hom(''P'', ''X''). If ''R'' is a ring, then in the category of ''R''-modules, a superfluous epimorphism is then an epimorphism p : P \to X such that the kernel of ''p'' is a superfluous submodule of ''P''. Properties Projective covers and their superfluous epimorphisms, when they exist, are unique up to isomorphism. The isomorphism need not be unique, however, since the projective property is not a full fledged universal property. The main effect of ''p'' having a superfluous kernel is the following: if ''N'' is any proper submodule of ''P'', then p(N) \n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of A Module
In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(''M'') of ''M''. Definition Let ''R'' be a ring and ''M'' a left ''R''-module. A submodule ''N'' of ''M'' is called maximal or cosimple if the quotient ''M''/''N'' is a simple module. The radical of the module ''M'' is the intersection of all maximal submodules of ''M'', :\mathrm(M) = \bigcap_ N Equivalently, :\mathrm(M) = \sum_ S These definitions have direct dual analogues for soc(''M''). Properties * In addition to the fact rad(''M'') is the sum of superfluous submodules, in a Noetherian module rad(''M'') itself is a superfluous submodule. * A ring for which rad(''M'') = for every right ''R''-module ''M'' is called a right V-ring. * For any module ''M'', rad(''M''/rad(''M'')) is zero. * ''M'' is a finitely generate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Socle (mathematics)
In mathematics, the term socle has several related meanings. Socle of a group In the context of group theory, the socle of a group ''G'', denoted soc(''G''), is the subgroup generated by the minimal normal subgroups of ''G''. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups. As an example, consider the cyclic group Z12 with generator ''u'', which has two minimal normal subgroups, one generated by ''u''4 (which gives a normal subgroup with 3 elements) and the other by ''u''6 (which gives a normal subgroup with 2 elements). Thus the socle of Z12 is the group generated by ''u''4 and ''u''6, which is just the group generated by ''u''2. The socle is a characteristic subgroup, and hence a normal subgroup. It is not necessarily transitively no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
