# nLab ascending chain condition

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 ascending chain of subobjects of $B$ is a direct 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 $i_n:A_{n} \hookrightarrow A_{n+1}$.

Given categories $C$ and $D$ with forgetful functor $F:C \to D$, an object $B$ is said to satisfy the ascending chain condition on subobjects of $F(B)$ if for every ascending 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 $i_n:A_{n} \hookrightarrow A_{n+1}$ is an isomorphism.

## 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 ascending chain condition on two-sided ideals if for every ascending 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 $i_n:A_{n} \hookrightarrow A_{n+1}$ 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 ascending chain condition on left ideals and right ideals for rings.