nLab essential ideal

Idea

Let $\mathcal{C}$ be a category of modules or bimodules over a fixed ring.

Definition

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, if $M$ has zero intersection with only the zero subobject of $N$.

Remark

(terminology) Some authors call an essential submodule $M \subset N$ a large submodule, or say that $N$ is an essential extension of $M$, while others argue that this should be called an inessential extension.

A monomorphism of modules $i \colon M\to N$ is called an 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, this terminology is also applied to ideals, i.e., submodules (or sub-bimodules, in the $2$-sided case) of the ring. Hence, one speaks of essential ideals, etc. For essential extensions, one considers extensions of algebras, where ‘essential’ still refers to non-zero 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.

Definition

A morphism $h$ in a class $H$ of morphisms in a category $\mathcal{C}$ is $H$-essential if, for every morphism $g$ in $\mathcal{C}$, $g$ is in $H$ whenever $g\circ h$ is in $H$.

For $H$ the class of monomorphisms in a category of modules, this reduces to the above notion of essential extensions.

Coessential epimorphisms are the dual notion; they define superfluous submodules.

The 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 of $R$.

Properties

Proposition

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.

Proposition

Every essential embedding $I\hookrightarrow M$ where $I$ is injective is an isomorphism.

See injective hull for related statements.

Proposition

A monomorphism $h\colon M\to N$ is essential iff, for all morphisms $g$, we have that $g$ is monic if $g\circ h$ is monic.

Proof

Indeed, if $h$ is essential and $g$ is not monic, then $Ker g\cap M\neq 0$, and hence $Ker (g|_M)\neq 0$ and $g\circ h$ is not monic. Conversely, suppose that $g$ is monic whenever $g\circ h$ 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.

Proposition

The socle of a module (the internal sum of all simple submodules) equals the intersection of all its essential submodules.

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.

• 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]

category: algebra, operator algebras