Essential extension

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

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

As a special case, an essential left ideal of ${\displaystyle R}$ is a left ideal that is essential as a submodule of the left module ${\displaystyle _{R}R}$. The left ideal has non-zero intersection with any non-zero left ideal of ${\displaystyle R}$. Analogously, an essential right ideal is exactly an essential submodule of the right ${\displaystyle R}$ module ${\displaystyle R_{R}}$.

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 ${\displaystyle N}$ is superfluous if for any other submodule ${\displaystyle H}$,

${\displaystyle N+H=M\,}$ 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)