- Pure submodule
In

mathematics , especially in the field ofmodule theory , the concept of**pure submodule**provides a generalization ofdirect summand , a type of particularly well-behaved piece of a module. Pure modules are complementary toflat module s and generalize Prüfer's notion ofpure subgroup s. While flat modules are those modules which leaveshort exact sequence s exact after tensoring, a pure submodule defines a short exact sequence that remains exact after tensoring with any module. Similarly a flat module is adirect limit s ofprojective module s, and a pure submodule defines a short exact sequence which is a direct limit ofsplit exact sequence s, each defined by a direct summand.**Definition**Let "R" be a ring, and let "M", "P" be

module s 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 ontensor product s "i"⊗id_{"X"}:"P"⊗"X" → "M"⊗"X" isinjective .Analogously, a

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:$sum\_\{j=1\}^n\; a\_\{ij\}x\_j\; =\; y\_i\; qquadmbox\{\; for\; \}\; i=1,ldots,m$then there also exist elements "x"_{1}',..., "x"_{"n"}'**in "P"**such that:$sum\_\{j=1\}^n\; a\_\{ij\}x\text{'}\_j\; =\; y\_i\; qquadmbox\{\; for\; \}\; i=1,ldots,m$**Examples**Every

subspace of avector space over a field is pure. Everydirect summand of "M" is pure in "M".A ring is von Neumann regular if and only if "every" submodule of "every" "R"-module is pure.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.**References***cite book | first = Tsit-Yuen | last = Lam | year = 1999 | title = Lectures on Modules and Rings | publisher = Springer | id = ISBN 0-387-98428-3

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