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,
implies that
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 RR.
The usual notations for essential extensions include the following two expressions:
(Lam 1999), and (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,
implies that .
The usual notations for superfluous submodules include:
(Lam 1999), and (Anderson & Fuller 1992)
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