In mathematics, specifically module theory, given a ring R and R-modules 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 which 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, and 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: (), and () 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: (), and ()