nLab descending chain condition

Definition

Given a category $C$ and an object $B:C$, let $\mathrm{Mono}(B)$ be the category whose objects are subobjects of $B$ and whose morphisms are monomorphisms. A descending chain of subobjects of $B$ is an inverse sequence in $\mathrm{Mono}(B)$, a sequence of subobjects $A:\mathbb{N} \to \mathrm{Mono}(B)$ with the following dependent sequence of monomorphisms: for natural number $n \in \mathbb{N}$, a dependent monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.

Given categories $C$ and $D$ with forgetful functor $F:C \to D$, an object $B$ is said to satisfy the descending chain condition on subobjects of $F(B)$ if for every descending chain of subobjects $(A, i_n)$ of $F(B)$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an isomorphism.

Subobjects (considered as isomorphism classes) form a possibly large partially ordered set (poset). A partially ordered set $(X,\geq)$ is satisfying a descending chain condition if any inverse sequence $x_1\geq x_2\geq x_3\geq\ldots$ eventually stabilizes, that is, there exists $n$ such that $x_n = x_{n+1} =x_{n+2}=\ldots$.

Examples

There is a forgetful functor $F:Ring \to BMod$ from the category Ring of rings to the category $BMod$ of bimodules, which forgets the multiplicative structure on the bimodule, that the canonical left action and right action of each ring $R$ have domain $R^2$ and are equal to each other and to the multiplicative binary operation of $R$, and that the canonical biaction of each ring $R$ has domain $R^3$.

Given a ring $R$, a two-sided ideal is a subobject of $F(R)$, a sub-$R$-$R$-bimodule. A ring $R$ is said to satisfy the descending chain condition on two-sided ideals if for every descending chain of two-sided ideals $(A, i_n)$ of $F(B)$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an isomorphism.

Similarly, there are forgetful functors $G:Ring \to LeftMod$ from $Ring$ to the category $LeftMod$ of left modules and $H:Ring \to RightMod$ from $Ring$ to the category $RightMod$ of right modules, and one could similarly define the descending chain condition on left ideals and right ideals for rings.

See also

• ascending chain condition

• Artinian bimodule

• Artinian ring

• Mittag-Leffler condition