essential ideal

Let $C$ be some category of modules or bimodules (say over a ring, algebra, or operator algebra). Then an subobject $M\subset N$ is an **essential submodule** of $N$ (or, we say that $N$ is an **essential extension** of $M$) if $M$ has nonzero intersection (pullback in more abstract situations) with any nonzero subobject of $N$ (or equivalently, $M$ has zero intersection with only the zero subobject? of $N$).

In particular, one applies this terminology to ideals, i.e. submodules (or subbimodules in the $2$-sided case) of a ring, algebra, or operator algebra itself. Hence we talk about **essential ideals**. For essential extensions, one considers extensions of algebras, where ‘essential’ still refers to non-intersection with submodules rather than with subalgebras.

category: algebra, operator algebras

Revised on September 13, 2013 20:39:28
by Toby Bartels
(98.23.131.69)