In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple if and only if every module over it is a semisimple module.
An indecomposable module is a module that is not a direct sum of two nonzero submodules. Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism rings, then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in group theory, is known as the Krull–Schmidt theorem.
A special case of a decomposition of a module is a decomposition of a ring: for example, a ring is semisimple if and only if it is a direct sum (in fact a product) of matrix rings over division rings (this observation is known as the Artin–Wedderburn theorem).
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spaces and Hilbert spaces. See the article decompositionofamodule for a way to write amodule as a direct sum of submodules. We give the construction first...
direct sum of simple modules. A direct sum decompositionofamodule into indecomposable modules is called an indecomposable decomposition. In many situations...
persistence module that admits adecomposition as a direct sum of interval modules is often simply called "interval decomposable." One of the primary...
considering the ring as amodule over itself, so that ideals are submodules. This also generalizes the primary decomposition form of the structure theorem...
Look up module or modular in Wiktionary, the free dictionary. Module, modular and modularity may refer to the concept of modularity. They may also refer...
modular decomposition is adecompositionofa graph into subsets of vertices called modules. Amodule is a generalization ofa connected component ofa graph...
programming is a software design technique that emphasizes separating the functionality ofa program into independent, interchangeable modules, such that...
context ofmodules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple...
Block might further decompose the spoken language module into modules for phonetic analysis and lexical forms: "Decomposition stops when all the components...
In abstract algebra, a uniserial module M is amodule over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any...
mathematics, the term cycle decomposition can mean: Cycle decomposition (graph theory), a partitioning of the vertices ofa graph into subsets, such that...
linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is adecompositionofa Hermitian, positive-definite...
by encapsulating the information into amodule or other construct which presents an interface. A common use of information hiding is to hide the physical...
Lasker–Noether primary decompositionof ideals in commutative Noetherian rings. Specifically, if an ideal J is decomposed as a finite intersection of primary ideals...
"Decompositionof persistence modules." Proceedings of the American Mathematical Society 148, no. 11 (2020): 4581-4596. Webb, Cary. "Decompositionof graded...
module is called a cyclic module if it is generated by one element. D A D-module is amodule over a ring of differential operators. decompositionA decomposition...
spaces. The spectral theorem also provides a canonical decomposition, called the spectral decomposition, of the underlying vector space on which the operator...
1972 paper On the Criteria to Be Used in Decomposing Systems into Modules, this dictum is expressed in terms of information hiding, and the terms cohesion...
reflects a minimal decompositionof the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since...
multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity; the primary decompositionof the module corresponds...
representation decomposes as a direct sum of irreducible representations, with each irreducible representation appearing in the decomposition with multiplicity...
there is still such adecompositionof the group algebra F[G] as a sum of blocks (one for each isomorphism type of simple module), but the situation is...
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