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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\, \ Equivalently, :\mathrm(M) = \sum\, \ 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. In fact, if ''M'' is finitely generated over a ring, then rad(''M'') itself is a superfluous submodule. This is because any proper submodule of ''M'' is contained in a maximal submodu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection (set Theory)
In set theory, the intersection of two Set (mathematics), sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is written using the symbol "\cap" between the terms; that is, in infix notation. For example: \\cap\=\ \\cap\=\varnothing \Z\cap\N=\N \\cap\N=\ The intersection of more than two sets (generalized intersection) can be written as: \bigcap_^n A_i which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols. Definition The intersection of two sets A and B, denoted by A \cap B, is the set of all objects that are members of both the sets A and B. In symbols: A \cap B = \. That is, x is an element of the intersection A \cap B if and only if x is both an element of A and an element of B. For example: * The intersection of the sets and is . * The n ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosocle
In mathematics, the term cosocle (socle meaning ''pedestal'' in French) has several related meanings. In group theory, a cosocle of a group ''G'', denoted by Cosoc(''G''), is the intersection of all maximal normal subgroups of ''G''. Adolfo Ballester-Bolinches, Luis M. Ezquerro, ''Classes of Finite Groups'', 2006, ,p. 97/ref> If ''G'' is a quasisimple group, then Cosoc(''G'') = Z(''G''). In the context of Lie algebras, a cosocle of a symmetric Lie algebra is the eigenspace of its structural automorphism that corresponds to the eigenvalue +1. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.) In the context of module theory, the cosocle of a module over a ring ''R'' is defined to be the maximal semisimple quotient of the module. See also * Socle *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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
V-ring (ring Theory)
In mathematics, a V-ring is a ring ''R'' such that every simple ''R''- module is injective. The following three conditions are equivalent: #Every simple left (respectively right) ''R''-module is injective. #The radical of every left (respectively right) ''R''-module is zero. #Every left (respectively right) ideal of ''R'' is an intersection of maximal left (respectively right) ideals of ''R''. A commutative ring is a V-ring if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ... it is Von Neumann regular. References Ring theory {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Module
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. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group. Definition The left ''R''-module ''M'' is finitely generated if there exist ''a''1, ''a''2, ..., ''a''''n'' in ''M'' such that for any ''x'' in ''M'', there exist ''r''1, ''r''2, ..., ''r''''n'' in ''R'' with ''x'' = ''r''1''a''1 + ''r''2''a''2 + ... + ''r''''n''''a''''n''. The set is referred to as a gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superfluous Submodule
In mathematics, specifically module theory, given a ring ''R'' and an ''R''- module ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of ''M'') if for every submodule ''H'' of ''M'', :H\cap N=\\, implies that H=\\, As a special case, an essential left ideal of ''R'' is a left ideal that is essential as a submodule of the left module ''R''''R''. The left ideal has non-zero intersection with any non-zero left ideal of ''R''. Analogously, an essential right ideal is exactly an essential submodule of the right ''R'' module ''R''''R''. The usual notations for essential extensions include the following two expressions: :N\subseteq_e M\, , and N\trianglelefteq M The dual notion of an essential submodule is that of superfluous submodule (or small submodule). A submodule ''N'' is superfluous if for any other submodule ''H'', :N+H=M\, implies that H=M\,. The usual notations for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Module
In 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. However, the property is named after Emmy Noether who was the first one to discover the true importance of the property. Characterizations and properties In the presence of the axiom of choice, two other characterizations are possible: *Any nonempty set ''S'' of submodules of the module has a maximal element (with respect to set inclusion). This is known as the maximum condition. *All of the submodules of the module are finitely generated. If ''M'' is a module and ''K'' a submodule, then ''M'' is Noetherian if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Module
In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cyclic submodule generated by a element of ''M'' equals ''M''. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring ''R''. Examples Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic groups of prime order. If ''I'' is a right ideal of ''R'', then ''I'' is simple as a right module if and only if ''I'' is a minimal non-zero right ideal: If ''M'' is a non-zero proper submodule of ''I'', then it is also a right ideal, so ''I'' is not minimal. Conversely, if ''I'' is not mini ... [...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 (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an 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 operations 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 scal ... [...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 the latter 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 equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
