nLab category of two-sided ideals in a ring

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Context

Algebra

Category theory

Contents

Idea

The collection of two-sided ideals in a ring, which happens to have a structure of a concrete category.

Definition

Given a ring RR in a universe 𝒰\mathcal{U}, the category of two-sided ideals in RR in 𝒰\mathcal{U} is a 𝒰 \mathcal{U} -large but locally 𝒰 \mathcal{U} -small 𝒰 \mathcal{U} -small-bicomplete concrete thin category TwoSidedIdeals 𝒰(R)\mathrm{TwoSidedIdeals}_\mathcal{U}(R) whose objects are the two-sided ideals in RR in 𝒰\mathcal{U}, the sub- R R - R R -bimodules of RR in 𝒰\mathcal{U}, and whose morphisms are the RR-RR-bimodule monomorphisms f:IJf:I \hookrightarrow J between two-sided ideals II and JJ in RR in 𝒰\mathcal{U}.

Properties

The category comes with a faithful functor F R:TwoSidedIdeals 𝒰(R)Bimod R,RF_{R}:\mathrm{TwoSidedIdeals}_\mathcal{U}(R) \to \mathrm{Bimod}_{R,R}.

Given a subset SRS \subseteq R with an injection i:SRi:S \hookrightarrow R, the two-sided ideal I SI_S generated by SS is defined as the initial two-sided ideal in RR with a function j:SI Sj:S \to I_S. This object always exists because TwoSidedIdeals 𝒰(R)\mathrm{TwoSidedIdeals}_\mathcal{U}(R) is 𝒰\mathcal{U}-small-complete, which means that the pullback of any 𝒰\mathcal{U}-small family of RR-RR-bimodule monomorphisms with codomain RR exists, and thus arbitrary intersections of two-sided ideals of RR exists. SS is called a subbase. If SS is a singleton, then I SI_S is called a principal two-sided ideal in RR.

The two-sided ideal generated by the empty set I I_\emptyset is the zero two-sided ideal, and is both the initial two-sided ideal in RR and the zero RR-RR-bimodule 00.

RR is the unit two-sided ideal or the improper two-sided ideal in RR, the terminal object in TwoSidedIdeals 𝒰(R)\mathrm{TwoSidedIdeals}_\mathcal{U}(R). A proper two-sided ideal is a two-sided ideal II with a function from the set IRI \cong R of RR-RR-bimodule isomorphisms between II and RR to the empty set f:(IR)f: (I \cong R) \to \emptyset.

See also

Last revised on May 25, 2022 at 20:15:27. See the history of this page for a list of all contributions to it.