nLab Noetherian bimodule

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Idea

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

Definition

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

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

Examples

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

α L(a,b)ab\alpha_L(a, b) \coloneqq a \cdot b
α R(a,b)ab\alpha_R(a, b) \coloneqq a \cdot b
α(a,b,c)abc\alpha(a, b, c) \coloneqq a \cdot b \cdot c

See also

Last revised on May 25, 2022 at 22:04:07. See the history of this page for a list of all contributions to it.