nLab essential ideal

Idea

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 $C^\ast$-algebraic setup the ideals in the definition are required to be closed and 2-sided.

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?.

A monomorphism of modules $i:M\to N$ is an essential embedding if $N$ is an essential extension of $i(M)$. This alternative terminology is useful to motivate the notion of essential embeddability in $N$ as a property of $M$.

If $R$ is a ring a (say, left) $R$-module $M\neq 0$ is uniform if every nonzero submodule of $M$ is essential. In other words, the intersection of any two nonzero submodules of $M$ is nonzero.

Literature

• wikipedia essential extension
• K. R. Goodearl, R. B. Warfield, An introduction to noncommutative Noetherian rings, London Math. Society Student Texts 16 (1st ed,), 1989, xviii+303 pp.; or 61 (2nd ed.), 2004, xxiv+344 pp.

In

• Jiří Adámek, Horst Herrlich, Jiří Rosický, Walter Tholen, Injective hulls are not natural, Algebra univers. 48 (2002) 379–388 doi

the following generalization is considered. A morphism h in a class $H$ of morphisms in a category $C$ is $H$-essential if for every morphism $g$, the composite $g\circ h$ lies in $H$ only if $g$ does. If $H$ is the class of monomorphisms we get the standard notion of an essential extension.

category: algebra, operator algebras