- Pure submodule
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 that remains exact after tensoring with any module. Similarly a flat module is a direct limits of projective modules, and a pure submodule defines a short exact sequence which is a direct limit of split exact sequences, each defined by a direct summand.
Let "R" be a ring, and let "M", "P" be
modules over "R". If "i": "P" → "M" is injective then "P" is a pure submodule of "M" if, for any "R"-module "X", the natural induced map on tensor products "i"⊗id"X":"P"⊗"X" → "M"⊗"X" is injective.
short exact sequence:of "R"-modules is pure exact if the sequence stays exact when tensored with any "R"-module "X". This is equivalent to saying that "f"("A") is a pure submodule of "B".
Purity can also be expressed element-wise; it is really a statement about the solvability of certain systems of linear equations. Specifically, "P" is pure in "M" if and only if the following condition holds: for any "m"-by-"n" matrix ("a""ij") with entries in "R", and any set "y"1,...,"y""m" of elements of "P", if there exist elements "x"1,...,"x""n" in "M" such that:then there also exist elements "x"1',..., "x""n"' in "P" such that:
If:is a short exact sequence with "B" being a
flat module, then the sequence is pure exact if and only if "C" is flat. From this one can deduce that pure submodules of flat modules are flat.
*cite book | first = Tsit-Yuen | last = Lam | year = 1999 | title = Lectures on Modules and Rings | publisher = Springer | id = ISBN 0-387-98428-3
Wikimedia Foundation. 2010.
Look at other dictionaries:
Pure subgroup — In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas.DefinitionA subgroup S of a… … Wikipedia
Singular submodule — In the branches of abstract algebra known as ring 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) ideals in R. In set notation it is… … Wikipedia
Injective module — In mathematics, especially in the area of 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. Specifically, if Q is a submodule of some… … Wikipedia
Flat module — In abstract algebra, a flat module over a ring R is an R module M such that taking the tensor product over R with M preserves exact sequences.Vector spaces over a field are flat modules. Free modules, or more generally projective modules, are… … Wikipedia
List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… … Wikipedia
List of abstract algebra topics — Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this … Wikipedia
Torsion (algebra) — In abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be free. Definition Let G be a group. An element g of G is called a torsion element if g has… … Wikipedia
Congruence lattice problem — In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most … Wikipedia
Von Neumann regular ring — In mathematics, a ring R is von Neumann regular if for every a in R there exists an x in R with : a = axa .One may think of x as a weak inverse of a ; note however that in general x is not uniquely determined by a .(The regular local rings of… … Wikipedia
Serial module — Chain ring redirects here. For the bicycle part, see Chainring. In abstract algebra, a uniserial module M is 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,… … Wikipedia