essential submodule

Let X be a submoduleMathworldPlanetmath of a module Y. We say that X is an essential submodule of Y, and that Y is an essential extension of X, if whenever A is a nonzero submodule of Y, then AX is also nonzero.

A monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath f:XY is an essential monomorphism if the image imf is an essential submodule of Y.

Title essential submodule
Canonical name EssentialSubmodule
Date of creation 2013-03-22 12:10:00
Last modified on 2013-03-22 12:10:00
Owner antizeus (11)
Last modified by antizeus (11)
Numerical id 6
Author antizeus (11)
Entry type Definition
Classification msc 16D80
Synonym essential extension
Synonym essential monomorphism
Related topic EssentialSubgroup
Related topic EssentialMonomorphisms
Related topic CompleteRingOfQuotientsOfReducedCommutativeRings
Related topic EpimorphicHull