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...

direct sum of simple modules. A direct sum decomposition of a module into indecomposable modules is called an indecomposable decomposition. In many situations...

decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID is a completely decomposable module...

considering the ring as a module over itself, so that ideals are submodules. This also generalizes the primary decomposition form of the structure theorem...

persistence module that admits a decomposition as a direct sum of interval modules is often simply called "interval decomposable." One of the primary...

modular decomposition is a decomposition of a graph into subsets of vertices called modules. A module is a generalization of a connected component of a graph...

spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module as a direct sum of submodules. We give the construction first...

connection to decomposition of R-modules. If M is an R-module and E = EndR(M) is its ring of endomorphisms, then A ⊕ B = M if and only if there is a unique idempotent...

maintain. Different types of decomposition are defined in computer sciences: In structured programming, algorithmic decomposition breaks a process down into well-defined...

Block might further decompose the spoken language module into modules for phonetic analysis and lexical forms: "Decomposition stops when all the components...

mathematics, the term cycle decomposition can mean: Cycle decomposition (graph theory), a partitioning of the vertices of a graph into subsets, such that...

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...

In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any...

aspect or "concern" of the desired functionality. A module interface expresses the elements that are provided and required by the module. The elements defined...

mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular...

reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since...

context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple...

see Decomposition of a module. For the proof of an important special case, see Simple Artinian ring. Since a finite-dimensional algebra over a field...

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)...

multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity; the primary decomposition of the module corresponds...

by encapsulating the information into a module or other construct which presents an interface. A common use of information hiding is to hide the physical...

"Decomposition of persistence modules." Proceedings of the American Mathematical Society 148, no. 11 (2020): 4581-4596. Webb, Cary. "Decomposition of graded...

denotes the highest weight module with highest weight λ {\displaystyle \lambda } . In the decomposition of V {\displaystyle V} , a certain isomorphism class...

linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite...

{\displaystyle R_{n}} is a two-sided ⁠ R 0 R_{0} ⁠-module, and the direct sum decomposition is a direct sum of ⁠ R 0 R_{0} ⁠-modules. R {\displaystyle R}...

Lasker–Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal J is decomposed as a finite intersection of primary ideals...

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...

also sometimes referred to as LR decomposition (factors into left and right triangular matrices). The LU decomposition was introduced by the Polish astronomer...

gives conditions for a Remak decomposition to exist and for its factors to be unique. If E ≠ 0 {\displaystyle E\neq 0} is a module that satisfies the ACC...

indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the...