Essential extension

In mathematics, specifically module theory, given a ring R and an R-module M with a submodule N, the module M is said to be an essential extension of N (or N is said to be an essential submodule or large submodule of M) if for every submodule H of M,

${\displaystyle H\cap N=\{0\}\,}$ implies that ${\displaystyle H=\{0\}\,}$

As a special case, an essential left ideal of R is a left ideal that is essential as a submodule of the left module _RR. The left ideal has non-zero intersection with any non-zero left ideal of R. Analogously, an essential right ideal is exactly an essential submodule of the right R module R_R.

The usual notations for essential extensions include the following two expressions:

${\displaystyle N\subseteq _{e}M\,}$ (Lam 1999), and ${\displaystyle N\trianglelefteq M}$ (Anderson & Fuller 1992)

The dual notion of an essential submodule is that of superfluous submodule (or small submodule). A submodule N is superfluous if for any other submodule H,

${\displaystyle N+H=M\,}$ implies that ${\displaystyle H=M\,}$.

The usual notations for superfluous submodules include:

${\displaystyle N\subseteq _{s}M\,}$ (Lam 1999), and ${\displaystyle N\ll M}$ (Anderson & Fuller 1992)