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...

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. pure submodule pure submodule Quillen–Suslin theorem The Quillen–Suslin theorem states that...

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...

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

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

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...

fractions K. An R-submodule M of a K-vector space V is a lattice if M is finitely generated over R. It is full if V = K · M. An R-submodule N of M that is...

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

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...

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...

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...

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...

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...

) {\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 ) = {...

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...

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...

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...

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...

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)....

function_alias_name; PHP: use module::submodule::symbol;, Rust: use module::submodule::{symbol1, symbol2};, Rust: use module::submodule::symbol as altname; Rust:...

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...

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...

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...

numba-jax. Discontinued developments include: pyMPI, pypar, MYMPI and the MPI submodule in ScientificPython. R bindings of MPI include Rmpi and pbdMPI, where...

{\displaystyle e_{1},\ldots ,e_{l}} is a basis of L. This allows identifying a submodule of L generated by g 1 , … , g k {\displaystyle g_{1},\ldots ,g_{k}} with...

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...

a number of independent converter submodules, each containing its own storage capacitor. The IGBTs in each submodule either bypass the capacitor or connect...

and h(G,B) = h(G,A)h(G,C) If A is finite then h(G,A) = 1. For A is a submodule of the G-module B of finite index, if either quotient is defined then...