nLab
essential ideal

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

In particular, one applies this terminology to ideals, i.e. submodules (or subbimodules in the 22-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.

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