Essential submodule

Summary

An essential submodule of a module $M$ is a submodule that meets every nonzero submodule of $M$ in a nonzero element. One then says that $M$ is an essential extension of it. Essential submodules are "large" in a lattice sense, even when they are small in size. Essential extensions are the basis of the theory of injective hulls, uniform dimension and Goldie's theorem.