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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...

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

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

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

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...

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

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...

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...

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

^{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...

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...

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...

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

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...

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

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...

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