Semisimple module, the Glossary
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.[1]
Table of Contents
49 relations: Abelian group, Abstract algebra, Artinian ring, Center (ring theory), Characteristic (algebra), Commutative ring, Direct product, Direct sum, Direct sum of modules, Division ring, Domain (ring theory), Endomorphism, Endomorphism ring, Exact sequence, Field (mathematics), Finite group, Finitely generated module, Graduate Texts in Mathematics, Group representation, Group ring, Homological algebra, Indecomposable module, Injective module, Jacobson radical, Kasch ring, Krull dimension, Maschke's theorem, Mathematics, Matrix ring, Maximal ideal, Module (mathematics), Noetherian ring, Projective module, Radical of a module, Reduced ring, Ring (mathematics), Ring homomorphism, Semiprimitive ring, Semisimple algebra, Semisimple module, Semisimple representation, Simple module, Socle (mathematics), Springer Science+Business Media, Subring, Vector space, Von Neumann regular ring, Wedderburn–Artin theorem, Weyl algebra.
Abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
See Semisimple module and Abelian group
Abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements.
See Semisimple module and Abstract algebra
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. Semisimple module and Artinian ring are ring theory.
See Semisimple module and Artinian ring
Center (ring theory)
In algebra, the center of a ring R is the subring consisting of the elements x such that for all elements y in R. It is a commutative ring and is denoted as Z(R); 'Z' stands for the German word Zentrum, meaning "center". Semisimple module and center (ring theory) are ring theory.
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Characteristic (algebra)
In mathematics, the characteristic of a ring, often denoted, is defined to be the smallest positive number of copies of the ring's multiplicative identity that will sum to the additive identity. Semisimple module and characteristic (algebra) are ring theory.
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Commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. Semisimple module and commutative ring are ring theory.
See Semisimple module and Commutative ring
Direct product
In mathematics, one can often define a direct product of objects already known, giving a new one.
See Semisimple module and Direct product
Direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics.
See Semisimple module and Direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. Semisimple module and direct sum of modules are module theory.
See Semisimple module and Direct sum of modules
Division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Semisimple module and division ring are ring theory.
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Domain (ring theory)
In algebra, a domain is a nonzero ring in which implies or. Semisimple module and domain (ring theory) are ring theory.
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Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself.
See Semisimple module and Endomorphism
Endomorphism ring
In mathematics, the endomorphisms of an abelian group X form a ring. Semisimple module and endomorphism ring are module theory and ring theory.
See Semisimple module and Endomorphism ring
Exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
See Semisimple module and Exact sequence
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. Semisimple module and field (mathematics) are ring theory.
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Finite group
In abstract algebra, a finite group is a group whose underlying set is finite.
See Semisimple module and Finite group
Finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. Semisimple module and finitely generated module are module theory.
See Semisimple module and Finitely generated module
Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
See Semisimple module and Graduate Texts in Mathematics
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication.
See Semisimple module and Group representation
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. Semisimple module and group ring are ring theory.
See Semisimple module and Group ring
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.
See Semisimple module and Homological algebra
Indecomposable module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Semisimple module and indecomposable module are module theory.
See Semisimple module and Indecomposable module
Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Semisimple module and injective module are module theory.
See Semisimple module and Injective module
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. Semisimple module and Jacobson radical are ring theory.
See Semisimple module and Jacobson radical
Kasch ring
In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring R for which every simple right R module is isomorphic to a right ideal of R. Analogously the notion of a left Kasch ring is defined, and the two properties are independent of each other. Semisimple module and Kasch ring are ring theory.
See Semisimple module and Kasch ring
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals.
See Semisimple module and Krull dimension
Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces.
See Semisimple module and Maschke's theorem
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
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Matrix ring
In 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. Semisimple module and matrix ring are ring theory.
See Semisimple module and Matrix ring
Maximal ideal
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. Semisimple module and maximal ideal are ring theory.
See Semisimple module and Maximal ideal
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. Semisimple module and module (mathematics) are module theory.
See Semisimple module and Module (mathematics)
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. Semisimple module and Noetherian ring are ring theory.
See Semisimple module and Noetherian ring
Projective module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Semisimple module and projective module are module theory.
See Semisimple module and Projective module
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. Semisimple module and radical of a module are module theory.
See Semisimple module and Radical of a module
Reduced ring
In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Semisimple module and reduced ring are ring theory.
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Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Semisimple module and ring (mathematics) are ring theory.
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Ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. Semisimple module and ring homomorphism are ring theory.
See Semisimple module and Ring homomorphism
Semiprimitive ring
In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. Semisimple module and semiprimitive ring are ring theory.
See Semisimple module and Semiprimitive ring
Semisimple algebra
In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical).
See Semisimple module and Semisimple algebra
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. Semisimple module and semisimple module are module theory and ring theory.
See Semisimple module and Semisimple module
Semisimple representation
In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations).
See Semisimple module and Semisimple representation
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. Semisimple module and simple module are module theory.
See Semisimple module and Simple module
Socle (mathematics)
In mathematics, the term socle has several related meanings. Semisimple module and socle (mathematics) are module theory.
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Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Subring
In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and that shares the same multiplicative identity as R. (Note that a subset of a ring R need not be a ring.) For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). Semisimple module and subring are ring theory.
See Semisimple module and Subring
Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.
See Semisimple module and Vector space
Von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring R (associative, with 1, not necessarily commutative) such that for every element a in R there exists an x in R with. Semisimple module and von Neumann regular ring are ring theory.
See Semisimple module and Von Neumann regular ring
Wedderburn–Artin theorem
In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras.
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Weyl algebra
Let (R,\Delta) be a partial differential ring with commuting derivatives \Delta. Semisimple module and Weyl algebra are ring theory.
See Semisimple module and Weyl algebra
References
[1] https://en.wikipedia.org/wiki/Semisimple_module
Also known as Completely reducible module, Semi-simple module, Semisimple local ring, Semisimple ring.