# 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 ${\rm im}f$ 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