Let $\mathcal{C}$ be category of modules or bimodules (say over a ring, algebra, or operator algebra).
A subobject $M\subset N$ is an essential submodule of $N$ if $M$ has non-zero intersection (pullback) with any non-zero subobject of $N$, or equivalently: $M$ has zero intersection with only the zero subobject of $N$.
(terminology) Some authors call an essential submodule $M \subset N$ a large submodule, or, say that $N$ is an essential extension of $M$; while other authors argue that this should be called in-essential extension.
A monomorphism of modules $i \colon M\to N$ is called essential monomorphism or an essential embedding if $N$ is an essential extension of $i(M)$. This alternative terminology is useful for motivating the notion of essential embeddability in $N$ as a property of $M$.
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 one speaks of essential ideals, etc. For essential extensions, one considers extensions of algebras, where ‘essential’ still refers to non-intersection with submodules rather than with subalgebras.
For $C^\ast$-algebras, essential ideals are required to be closed and 2-sided.
In AHRT02 the following generalized notion is considered
A morphism $h$ among a class $H$ of morphisms in a category $\mathcal{C}$ is $H$-essential if for every morphism $g$ in $\mathcal{C}$, the composite $g\circ h$ lies in $H$ only if $g$ does.
Coessential epimorphisms are the dual notion; they define superfluous submodules.
Singular submodule of a left $R$-module $M$ is the subset $\mathcal{Z}(M)$ of all elements in $M$ whose annihilator is an essential left ideal in $R$.
If $R$ is a ring, a (left, say) $R$-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.
Every essential embedding $I\hookrightarrow M$ where $I$ is injective is an isomorphism.
A monomorphism $h\colon M\to N$ is essential iff $g\circ h$ is monic only if $g$ is monic.
Indeed, if $h$ is essential and $g$ is not monic, then $Ker g\cap M\neq 0$ hence $Ker (g|_M)\neq 0$ and $g\circ h$ is not monic. Conversely, suppose $g\circ h$ is monic implies $g$ is monic. If $h$ were not essential then there would be $0\neq K\subset N$ such that $K\cap M = 0$; thus the cokernel map $g: N\to N/K$ is not monic while $g\circ h$ is monic because $Ker(g)\cap Im(h) = K\cap M = 0$ and $h$ is monic. Since this is a contradiction, the claim follows.
The socle of a module (internal sum of all simple submodules) equals the intersection of all essential submodules.
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.
Louis H. Rowen, starting from Def. 2.10.16 in: Ring theory – Student edition, Academic Press (1991, 2012)
Jiří Adámek, Horst Herrlich, Jiří Rosický, Walter Tholen, Injective hulls are not natural, Algebra univers. 48 (2002) 379-388 [doi:10.1007/s000120200006]
Last revised on August 20, 2024 at 16:09:31. See the history of this page for a list of all contributions to it.