essential submodule

Let $X$ be a submodule 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 $A\cap X$ is also nonzero.

A monomorphism $f:X\to Y$ is an essential monomorphism if the image $\mathrm{im}f$ 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