nLab essential ideal

Redirected from "essential monomorphisms".

Contents

Idea

Let 𝒞\mathcal{C} be category of modules or bimodules (say over a ring, algebra, or operator algebra).

Definition

A subobject MNM\subset N is an essential submodule of NN if MM has non-zero intersection (pullback) with any non-zero subobject of NN, or equivalently: MM has zero intersection with only the zero subobject of NN.

Remark

(terminology) Some authors call an essential submodule MNM \subset N a large submodule, or, say that NN is an essential extension of MM; while other authors argue that this should be called in-essential extension.

A monomorphism of modules i:MNi \colon M\to N is called essential monomorphism or an essential embedding if NN is an essential extension of i(M)i(M). This alternative terminology is useful for motivating the notion of essential embeddability in NN as a property of MM.

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 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 * C^\ast -algebras, essential ideals are required to be closed and 2-sided.

In AHRT02 the following generalized notion is considered

Definition

A morphism hh among a class HH of morphisms in a category 𝒞\mathcal{C} is HH-essential if for every morphism gg in 𝒞\mathcal{C}, the composite ghg\circ h lies in HH only if gg does.

For HH is the class of monomorphisms in a category of modules this reduces again to the above notion of essential extension.

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

Singular submodule of a left RR-module MM is the subset 𝒵(M)\mathcal{Z}(M) of all elements in MM whose annihilator is an essential left ideal in RR.

Properties

Proposition

If RR is a ring, a (left, say) RR-module M0M\neq 0 is uniform if every nonzero submodule of MM is essential. In other words, the intersection of any two nonzero submodules of MM is nonzero.

Proposition

Every essential embedding IMI\hookrightarrow M where II is injective is an isomorphism.

See also injective hull for related statements.

Proposition

A monomorphism h:MNh\colon M\to N is essential iff ghg\circ h is monic only if gg is monic.

Proof

Indeed, if hh is essential and gg is not monic, then KergM0Ker g\cap M\neq 0 hence Ker(g| M)0Ker (g|_M)\neq 0 and ghg\circ h is not monic. Conversely, suppose ghg\circ h is monic implies gg is monic. If hh were not essential then there would be 0KN0\neq K\subset N such that KM=0K\cap M = 0; thus the cokernel map g:NN/Kg: N\to N/K is not monic while ghg\circ h is monic because Ker(g)Im(h)=KM=0Ker(g)\cap Im(h) = K\cap M = 0 and hh is monic. Since this is a contradiction, the claim follows.

Proposition

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

Literature

Last revised on August 20, 2024 at 16:09:31. See the history of this page for a list of all contributions to it.