nLab Noetherian bimodule

Idea

A Noetherian bimodule is a bimodule which satisfies the ascending chain condition on its subbimodules.

Definition

Given rings $R$ and $S$ and an $R$-$S$-bimodule $B$, let $\mathrm{Mono}(B)$ be the category whose objects are $R$-$S$-subbimodules of $B$ and whose morphisms are $R$-$S$-bimodule monomorphisms. An ascending chain of $R$-$S$-subbimodules is a direct sequence of $R$-$S$-subbimodules in $\mathrm{Mono}(B)$, a sequence of $R$-$S$-subbimodules $A:\mathbb{N} \to \mathrm{Mono}(B)$ with the following dependent sequence of $R$-$S$-bimodule monomorphisms: for natural number $n \in \mathbb{N}$, a dependent $R$-$S$-bimodule monomorphism $i_n:A_n \hookrightarrow A_{n+1}$.

An $R$-$S$-bimodule $B$ is Noetherian if it satisfies the ascending chain condition on its subbimodules: for every ascending chain of $R$-$S$-subbimodules $(A, i_n)$ of $B$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the $R$-$S$-bimodule monomorphism $i_n:A_n \hookrightarrow A_{n+1}$ is an $R$-$S$-bimodule isomorphism.

Examples

A ring $R$ is Noetherian if it is Noetherian as a $R$-$R$-bimodule with respect to its canonical bimodule structure, with its left action $\alpha_L:R \times R \to R$ and right action $\alpha_R:R \times R \to R$ defined as its multiplicative binary operation and its biaction $\alpha:R \times R \times R \to R$ defined as its ternary product:

$$\alpha_L(a, b) \coloneqq a \cdot b$$

$$\alpha_R(a, b) \coloneqq a \cdot b$$

$$\alpha(a, b, c) \coloneqq a \cdot b \cdot c$$

See also

• bimodule

• Artinian bimodule

• Noetherian ring