nLab ascending chain condition

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Definition

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

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

Examples

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

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

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

See also

Created on May 25, 2022 at 21:11:43. See the history of this page for a list of all contributions to it.