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. Equivalently, a module M...

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...

trivial module 0 is semi-simple, but not simple). For an R-module M, M is semi-simple if and only if it is the direct sum of simple modules (the trivial...

R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that...

the number of simple modules k(G) is equal to the number of conjugacy classes of G. In the modular case, the number l(G) of simple modules is equal to the...

Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple means "no proper submodule" N < M, while...

and the vector space M = N is a simple module of R. Then Schur's lemma says that the endomorphism ring of the module M is a division algebra over k. If...

structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that...

intersection theory. The zero module is the only one with length 0. Modules with length 1 are precisely the simple modules. The length of the cyclic group...

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...

two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a...

analogous to an irreducible representation in the resulting theory is a simple module.[citation needed] Let ρ {\displaystyle \rho } be a representation i...

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...

arises in this fashion from some simple module. Much of linear algebra may be formulated, and remains correct, for modules over a division ring D instead...

subgroup. A ring is called a simple ring if it does not contain a nontrivial two sided ideal. A module is called a simple module if it does not contain a...

those elements in R {\displaystyle R} that annihilate all simple right R {\displaystyle R} -modules. It happens that substituting "left" in place of "right"...

algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring...

ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R. The theorem can be...

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

{g}}W} is simple (resp. absolutely simple), then W is simple (resp. absolutely simple). Here, a g {\displaystyle {\mathfrak {g}}} -module V is absolutely...

theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive...

finite-dimensional module over g {\displaystyle {\mathfrak {g}}} is semisimple as a module (i.e., a direct sum of simple modules.) Weyl's theorem implies...

hold when the image of K[G] is a simple algebra, with centre K (by what is called Schur's lemma: see simple module). As a consequence, in important cases...

proper subrepresentations. Similarly, an irreducible module is another name for a simple module. Absolutely irreducible is a term applied to mean irreducible...

A semisimple module is a direct sum of simple modules. A semisimple ring is a ring that is semisimple as a left module (or right module) over itself....

non-trivial two-sided ideals is called quasi-simple. Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring...

The Apollo command and service module (CSM) was one of two principal components of the United States Apollo spacecraft, used for the Apollo program, which...

A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible)...

mod 2 {\displaystyle \ell \equiv n{\bmod {2}}} . The dimension of a simple module is written in terms of binomial coefficients as dim ⁡ ( W ℓ ) = ( n...

device that converts sunlight into electricity by using multiple solar modules that consist of photovoltaic (PV) cells. PV cells are made of materials...