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...

ring. The torsion submodule of a module is the submodule formed by the torsion elements (in cases when this is indeed a submodule, such as when the ring...

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...

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...

(ring theory) Structure Submodule Pure submodule Module homomorphism Essential submodule Superfluous submodule Singular submodule Socle of a module Radical...

Nagata ring. pure 1.  A pure submodule M of a module N is a submodule such that M⊗A is a submodule of N⊗A for all modules A. 2.  A pure subring R of a...

projective cover is a minimal surjection from a projective module. pure submodule pure submodule Quillen–Suslin theorem The Quillen–Suslin theorem states that...

consideration of pure subgroups. Pure subgroups were generalized in several ways in the theory of abelian groups and modules. Pure submodules were defined...

and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left)...

R-submodule N of M that is itself a lattice is an R-pure sublattice if M/N is R-torsion-free. There is a one-to-one correspondence between R-pure sublattices...

fellow of the American Mathematical Society. Zimmermann-Huisgen, Birge: Pure submodules of direct products of free modules. Math. Ann. 224 (1976), no. 3, 233–245...

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...

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)....

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...

is flat. Every ideal of R is flat Every overring of R is R-flat Every submodule of a flat R-module is flat. If M and N are torsion-free R-modules then...

R-module M is simple, if it has no submodules other than 0 and M. An R-module M is semi-simple if every R-submodule of M is an R-module direct summand...

– C++ use module::submodule::symbol; – Rust use module::submodule::{symbol1, symbol2}; – Rust use module::submodule::symbol as altname; – Rust...

every module A over a PID has a free resolution of length 1, since every submodule of a free module over a PID is free. For any ring R, Tor preserves direct...

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...

) {\displaystyle C^{p}(X)} consisting of locally zero functions is a submodule, denote by C 0 p ( X ) {\displaystyle C_{0}^{p}(X)} . C 0 ∗ ( X ) = {...

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...

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...

3.1) was released four days later to address a Git vulnerability with submodules (CVE 2017-1000117). On October 10, 2017, Visual Studio 15.4 was released...

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...

constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc. A simple...

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...

Instead of taking an irreducible quotient, one can take an irreducible submodule. Since tempered representations are in turn given as certain representations...

ring. Von Neumann algebras are semihereditary: every finitely generated submodule of a projective module is itself projective. There have been several attempts...