symmetric monoidal (∞,1)-category of spectra
An Artinian bimodule is a bimodule which satisfies the descending chain condition on its subbimodules.
Given rings and and an --bimodule , let be the category whose objects are --subbimodules of and whose morphisms are --bimodule monomorphisms. A descending chain of --subbimodules is an inverse sequence of --subbimodules in , a sequence of --subbimodules with the following dependent sequence of --bimodule monomorphisms: for natural number , a dependent --bimodule monomorphism .
An --bimodule is Artinian if it satisfies the descending chain condition on its subbimodules: for every descending chain of --subbimodules of , there exists a natural number such that for all natural numbers , the --bimodule monomorphism is an --bimodule isomorphism.
A ring is Artinian if it is Artinian as a --bimodule with respect to its canonical bimodule structure, with its left action and right action defined as its multiplicative binary operation and its biaction defined as its ternary product:
Last revised on May 26, 2022 at 02:03:52. See the history of this page for a list of all contributions to it.