Let be some category of modules or bimodules (say over a ring, algebra, or operator algebra). Then an subobject is an essential submodule of (or, we say that is an essential extension of ) if has nonzero intersection (pullback in more abstract situations) with any nonzero subobject of (or equivalently, has zero intersection with only the zero subobject? of ).
In particular, one applies this terminology to ideals, i.e. submodules (or subbimodules in the -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?.
Last revised on September 13, 2013 at 20:39:28. See the history of this page for a list of all contributions to it.