Module components with flexibility in module theory
In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modules which leave short exact sequences exact after tensoring, a pure submodule defines a short exact sequence (known as a pure exact sequence) that remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure exact sequence is a direct limit of split exact sequences.
the concept of puresubmodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary...
Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module...
projective cover is a minimal surjection from a projective module. puresubmodulepuresubmodule Quillen–Suslin theorem The Quillen–Suslin theorem states that...
M with a submodule N, the module M is said to be an essential extension of N (or N is said to be an essential submodule or large submodule of M) if for...
and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left)...
consideration of pure subgroups. Pure subgroups were generalized in several ways in the theory of abelian groups and modules. Puresubmodules were defined...
direct sum of cyclic groups of order pn and infinite cyclic groups; B is a p-pure subgroup of A; The quotient group, A/B, is a p-divisible group. Conditions...
injective module B, then E(M) is (isomorphic to) a submodule of B. If N is an essential submodule of M, then E(N)=E(M). Every module M has an injective...
of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of M is an essential submodule. A ring may be called...
subrepresentation, such that the inertia group acts in a certain way on the submodule and the quotient. The exact condition depends on the author; for example...
examples, suppose S is some ring and M is an S-module such that every submodule of M is a direct summand of M (such modules M are called semisimple)....
such that every submodule of a module of finite type is also of finite type. Ideals of a ring R {\displaystyle R} are the submodules of R {\displaystyle...
where M tor {\displaystyle M_{\operatorname {tor} }} is the torsion submodule of M. If M is a torsion R-module then K ⊗ R M = 0 {\displaystyle K\otimes...
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...
implementations in Python include: pyMPI, mpi4py, pypar, MYMPI, and the MPI submodule in ScientificPython. pyMPI is notable because it is a variant python interpreter...
finitely generated free R-module, and for every finitely generated R-submodule K {\displaystyle K} of ker f , {\displaystyle \ker f,} the map f {\displaystyle...
a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either N 1 ⊆ N...
superalgebra. Let A be a superalgebra over a commutative ring K. The submodule A0, consisting of all even elements, is closed under multiplication and...
ring. Von Neumann algebras are semihereditary: every finitely generated submodule of a projective module is itself projective. There have been several attempts...
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