abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Indecomposable is a weaker notion...

simple module is indecomposable, but there are indecomposable modules that are not simple (e.g. uniform modules). Faithful A faithful module M is one...

as module theory, a principal indecomposable module has many important relations to the study of a ring's modules, especially its simple modules, projective...

mathematics: Indecomposable module, in algebra Indecomposable distribution, in probability Indecomposable continuum, in topology Indecomposability (intuitionistic...

every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module over one ring may...

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...

R/(q_{i})} are indecomposable, so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID...

module over itself and a module over Z. The rational numbers are algebraically compact as a Z-module. Together with the indecomposable finite modules...

The simple modules are precisely the modules of length 1; this is a reformulation of the definition. Every simple module is indecomposable, but the converse...

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...

Localization of a module Completion (ring theory) Types Simple module, Semisimple module Indecomposable module Artinian module, Noetherian module Homological...

indecomposable An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is...

^{n}} and R n . {\displaystyle \mathbb {R} ^{n}.} The direct product for modules (not to be confused with the tensor product) is very similar to the one...

This allows them to derive relativistic wave equations. Simple module Indecomposable module Representation of an associative algebra Representation theory...

combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with...

maximal submodule, then M is a local module. M is also clearly a uniform module and thus is directly indecomposable. It is also easy to see that every finitely...

algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of...

modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of indecomposable injective modules...

basic statement in abstract algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every endomorphism of M...

algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a...

regular module). Each projective indecomposable module (and hence each projective module) in positive characteristic p may be lifted to a module in characteristic...

Theorem (Lam 2001, §19): A module is said to be strongly indecomposable if its endomorphism ring is a local ring. For a module M {\displaystyle M} of finite...

N_{2}/(N_{1}\cap N_{2})=\{0\}.} Uniserial modules are uniform, and uniform modules are necessarily directly indecomposable. Any commutative domain is a uniform...

finite-dimensional algebras over k of global dimension ≤ 2 such that every indecomposable module either has projective dimension ≤ 1 or injective dimension ≤ 1....

coprimary modules. For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable injective modules onto the...

If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring. For a semisimple module, the endomorphism...

idempotent of a ring R is a nonzero idempotent a such that aR is indecomposable as a right R-module; that is, such that aR is not a direct sum of two nonzero...

then E {\displaystyle E} is a direct sum of indecomposable modules. Up to a permutation, the indecomposable components in such a direct sum are uniquely...

defined for modules over any integral domain, the case of abelian groups corresponding to modules over Z. For this, see finitely generated module#Generic...

modules over some other rings. Specifically, if the endomorphism ring of the module M is local, then M is indecomposable; conversely, if the module M...