Definition
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⊗idX:P⊗X → M⊗X is injective.
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 (aij) with entries in R, and any set y1,...,ym of elements of P, if there exist elements x1,...,xn in M such that
then there also exist elements x1',..., xn' in P such that
Read more about this topic: Pure Submodule
Famous quotes containing the word definition:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)