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Module components with flexibility in 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
Pure_submodule
Mathematical object in abstract algebra
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
Injective_module
Zero divisors in a module
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
Torsion_(algebra)
Concept in mathematics
essential submodule is that of superfluous submodule (or small submodule). A submodule N {\displaystyle N} is superfluous if for any other submodule H {\displaystyle
Essential_extension
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
Glossary of commutative algebra
Glossary_of_commutative_algebra
projective cover is a minimal surjection from a projective module. pure submodule pure submodule Quillen–Suslin theorem The Quillen–Suslin theorem states that
Glossary_of_module_theory
and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left)
Singular_submodule
Branch of mathematics that studies algebraic structures
Annihilator (ring theory) Submodule Pure submodule Module homomorphism Essential submodule Superfluous submodule Singular submodule Socle of a module Radical
List of abstract algebra topics
List_of_abstract_algebra_topics
consideration of pure subgroups. Pure subgroups were generalized in several ways in the theory of abelian groups and modules. Pure submodules were defined
Pure_subgroup
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
Lattice_(module)
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
Basic_subgroup
American mathematician
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
Birge_Huisgen-Zimmermann
Notion in abstract algebra
{\displaystyle E(M)} is (isomorphic to) a submodule of B {\displaystyle B} . If N {\displaystyle N} is an essential submodule of M {\displaystyle M} , then E (
Injective_hull
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
Prüfer_domain
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
Uniform_module
Rings admitting weak inverses
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).
Von_Neumann_regular_ring
Mathematical terminology
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
Galois_representation
Algebra with unique prime factorization
order of the factors. (M2) The torsion submodule is a direct summand. That is, there exists a complementary submodule P {\displaystyle P} of M {\displaystyle
Dedekind_domain
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
Serial_module
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
Herbrand_quotient
Mathematical property
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
Semi-simplicity
Abelian group in which every element can, in some sense, be divided by positive integers
quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective. The converse is a result
Divisible_group
Abstract algebra concept
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
Decomposition_of_a_module
Cohomology theory for topological spaces
) {\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 ) = {
Alexander–Spanier_cohomology
original on 2014-02-02. Retrieved 2014-01-26. "FossilHelp: import" "git-submodule(1) Manual Page". Kernel.org. 2013-02-15. Retrieved 2014-01-26. "git-read-tree(1)
Comparison of version-control software
Comparison_of_version-control_software
Construction in homological algebra
{\displaystyle 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 {\displaystyle R}
Tor_functor
Operation that pairs a left and a right R-module into an abelian group
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
Tensor_product_of_modules
– C++ use module::submodule::symbol; – Rust use module::submodule::{symbol1, symbol2}; – Rust use module::submodule::symbol as altname; – Rust
Comparison of programming languages (syntax)
Comparison_of_programming_languages_(syntax)
Integrated development environment by Microsoft
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
Visual_Studio
Algebraic structure in ring theory
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
Flat_module
German mathematician (1882–1935)
Noetherian module is a module in which every strictly ascending chain of submodules becomes constant after a finite number of steps. A Noetherian space is
Emmy_Noether
Algebraic structure
independents. A module that has a basis is called a free module, and a submodule of a free module needs not to be free. A module of finite type is a module
Commutative_ring
Important problem in lattice theory
generated congruences below it (e.g., every submodule of a module is the union of all its finitely generated submodules), we obtain the following result, first
Congruence_lattice_problem
Mathematical theory
Instead of taking an irreducible quotient, one can take an irreducible submodule. Since tempered representations are in turn given as certain representations
Langlands_classification
Writing Lie algebra sets as matrices
constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc. A simple
Lie_algebra_representation
Electric power transmission system
a number of independent converter submodules, each containing its own storage capacitor. The IGBTs in each submodule either bypass the capacitor or connect
High-voltage_direct_current
French mathematician
named after Demazure, who wrote about them in 1974. Demazure modules are submodules of a finite-dimensional representation of a semisimple Lie algebra, and
Michel_Demazure
Message-passing system for parallel computers
numba-jax. Discontinued developments include: pyMPI, pypar, MYMPI and the MPI submodule in ScientificPython. R bindings of MPI include Rmpi and pbdMPI, where
Message_Passing_Interface
*-algebra of bounded operators on a Hilbert space
ring. Von Neumann algebras are semihereditary: every finitely generated submodule of a projective module is itself projective. There have been several attempts
Von_Neumann_algebra
Algebraic structure used in theoretical physics
A} be a superalgebra over a commutative ring K {\displaystyle K} . The submodule A 0 {\displaystyle A_{0}} , consisting of all even elements, is closed
Superalgebra
allows for the module-theoretic terminology: e.g., trivial G-module, G-submodules, etc. G-equivariant vector bundle A G-equivariant vector bundle is a vector
Glossary of representation theory
Glossary_of_representation_theory
Mathematical construct in computer algebra
{\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
Gröbner_basis
Special subset of a partially ordered set
Cartan 1937b. Bumby, R. T. (1965-12-01). "Modules which are isomorphic to submodules of each other". Archiv der Mathematik. 16 (1): 184–185. doi:10.1007/BF01220018
Filter_(mathematics)
Algebra of formal sums
generalized to free modules over a principal ideal domain. For instance, submodules of free modules over principal ideal domains are free, a fact that Hatcher
Free_abelian_group
Intrinsic geometric structures in mathematics
Riemannian metric on M. Since X {\displaystyle {\mathcal {X}}} (M) is a submodule of C∞(M, E3)=C∞(M) ⊗ {\displaystyle \otimes } E3, the operator X ⊗ {\displaystyle
Riemannian connection on a surface
Riemannian_connection_on_a_surface
PURE SUBMODULE
PURE SUBMODULE
Boy/Male
Hindu, Indian
Pure; Sure
Girl/Female
Greek
Soft breeze.
Surname or Lastname
Scottish and Irish
Scottish and Irish : reduced form of McCure, an Anglicized form of Gaelic Mac Ãomhair (see McIver).English : possibly from Middle English cure ‘charge’, ‘care’, ‘concern’.
Boy/Male
Australian, Greek
Farmer
Male
Croatian
, farmer, husbandman.
Girl/Female
English
Prudence. One of the many qualities and virtues that the Puritans adopted as names after the...
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Heaven; Abundant; Name of a King; A Legendary King Yayati
Surname or Lastname
English
English : metonymic occupational name for someone who made bags or purses or for an official in charge of expenditure, from Middle English purse (via Old English from Latin bursa).Scottish : variant of Purser.
Boy/Male
Australian, French, Portuguese
Stern; Severe
Surname or Lastname
English
English : nickname for a redoubtable warrior, from Middle English prou(s) ‘brave’, ‘valiant’ (Old French proux, preux).Americanized spelling of French Prou (see Proulx).
Male
Swedish
Danish and Swedish form of Scandinavian Tore, TURE means "thunder."
Male
Slovene
Croatian and Slovene form of Greek Georgios, JURE means "earth-worker, farmer."
Boy/Male
Hindu
Unique, Unmatched, New
Boy/Male
Hindu
Abundant, Name of a king
Boy/Male
Gujarati, Indian
In Begining
Girl/Female
Hindu
City
Girl/Female
Spanish
Pure.
Surname or Lastname
Southern French (Péré)
Southern French (Péré) : topographic name from a variant of périer ‘pear tree’.Catalan : from the personal name Pere, Catalan equivalent of Peter.English : variant of Pear 1.Hungarian : from the old secular personal name Pere, Pöre.
Male
Ukrainian
, farmer.
Boy/Male
Australian, Hindu, Indian
Devotional Place
PURE SUBMODULE
PURE SUBMODULE
Boy/Male
Indian, Punjabi, Sikh
Eternal Abode
Girl/Female
Muslim
Glorious, Sacred
Boy/Male
Muslim/Islamic
Grasshopper (name of a companion)
Girl/Female
Tamil
Alvira | அலà¯à®µà¯€à®°à®¾
Speaker of truth
Girl/Female
Hindu, Indian, Telugu
Gift of God
Girl/Female
Tamil
Stars
Boy/Male
Hindu, Indian
A Raga
Girl/Female
Afghan, African, Arabic, Swahili
Beloved
Girl/Female
Gujarati, Hindu, Indian
Goddess Laxmi; Beautiful Women
Surname or Lastname
Irish
Irish : Anglicized form of Gaelic Ó Baoighealláin. It was the name of a sept of Dartry, County Monaghan.English : variant of Boyland.
PURE SUBMODULE
PURE SUBMODULE
PURE SUBMODULE
PURE SUBMODULE
PURE SUBMODULE
v. t.
To cut off, or shave off, the superficial substance or extremities of; as, to pare an apple; to pare a horse's hoof.
adv.
In a sure manner; safely; certainly.
a.
Pure; undefiled.
v. i.
To restore health; to effect a cure.
n.
Hence, a treasury; finances; as, the public purse.
v. i.
To become pure, as by clarification.
n.
The low murmuring sound made by a cat; pur. See Pur.
superl.
Separate from all heterogeneous or extraneous matter; free from mixture or combination; clean; mere; simple; unmixed; as, pure water; pure clay; pure air; pure compassion.
v. t.
To remove; to separate; to cut or shave, as the skin, ring, or outside part, from anything; -- followed by off or away; as; to pare off the ring of fruit; to pare away redundancies.
v. t.
To put into a purse.
v. t.
To draw up or contract into folds or wrinkles, like the mouth of a purse; to pucker; to knit.
n.
Spiritual charge; care of soul; the office of a parish priest or of a curate; hence, that which is committed to the charge of a parish priest or of a curate; a curacy; as, to resign a cure; to obtain a cure.
n.
A gentle murmur, as that produced by the running of a liquid among obstructions; as, the purl of a brook.
n.
Medical or hygienic care; remedial treatment of disease; a method of medical treatment; as, to use the water cure.
v. t.
To clear from guilt, or from moral or ceremonial defilement; as, to purge one of guilt or crime.
superl.
Certain to find or retain; as, to be sure of game; to be sure of success; to be sure of life or health.
v. i. & t.
To murmur as a cat. See Pur.
n.
To draw to the lure; hence, to allure or invite by means of anything that promises pleasure or advantage; to entice; to attract.
v. t.
To prepare for preservation or permanent keeping; to preserve, as by drying, salting, etc.; as, to cure beef or fish; to cure hay.
n.
A sum of money offered as a prize, or collected as a present; as, to win the purse; to make up a purse.