# Artinian ring

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## Artinian ring

[ar¦tin·ē·ən ′riŋ]
(mathematics)
A ring is Artinian on left ideals (or right ideals) if every descending sequence of left ideals (or right ideals) has only a finite number of distinct members.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
Suppose A is an artinian ring, with indecomposable A-modules {[P.sub.[alpha]] |[alpha] [member of] I} (representatives from each isomorphism class for some index set I).
Suppose R is an artinian ring that is not semisimple and with two additional indecomposable modules [I.sub.1], [I.sub.2] that are not projective and not isomorphic.
Every finitely generated module over a right Artinian ring is [pi]-Rickart (see Theorem 2.30), every free module which its endomorphism ring is generalized right principally projective is [pi]-Rickart (see Corollary 3.5), every finitely generated projective regular module is [pi]-Rickart (see Corollary 3.7) and every finitely generated projective module over a commutative [pi]-regular ring is [pi]-Rickart (see Proposition 3.11).
Let R be a right Artinian ring and M a finitely generated R-module.
Every right Artinian ring is generalized right principally projective.
Let R be a right Artinian ring. Then [M.sub.n] (R) is generalized right principally projective for every positive integer n.
They are orders in Artinian rings, Goldie's theorem, and the largest left quotient ring of a ring; the invariant theory of Artin-Schelter regular algebras: a survey; the Catalan combinatorics of the hereditary Artin algebras; Grassmannians, flag varieties, and Gelfand-Zetlin polytopes; and on the combinatorics of the set of tilting modules.
of Wisconsin- Madison) also includes coverage of rarer subjects in this field, including transcendental field extensions, modules over Dedekind domains and artinian rings. Unlike similar textbooks, this volume steers away from chapter-end problems by including full details of all proofs as problems are presented.
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