nLab category of two-sided ideals in a ring

Idea

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

Definition

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

Properties

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

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

The two-sided ideal generated by the empty set $I_\emptyset$ is the zero two-sided ideal, and is both the initial two-sided ideal in $R$ and the zero $R$-$R$-bimodule $0$.

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

See also

• ring

• bimodule

• two-sided ideal

• category of subobjects