About: Goldie's theorem

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In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (="finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem.

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rdfs:label	Goldie's theorem (en)
rdfs:comment	In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (="finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. (en)
dcterms:subject	Theorems in ring theory
Wikipage page ID	7738492 (xsd:integer)
Wikipage revision ID	1096489996 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime ring Semiprime ring Nonsingular ring Alfred Goldie Integral domain Artin–Wedderburn theorem Mathematics Nil ideal Noetherian ring Annihilator (ring theory) Matrix ring Ring (mathematics) Ring theory Artinian ring Theorems in ring theory Ascending chain condition Semisimple ring Springer-Verlag Ore domain Ore ring Classical ring of quotients Regular element (ring theory) Principal right ideal ring Essential submodule Uniform dimension
Link from a Wikipage to an external page	http://planetmath.org/encyclopedia/GoldieRing.html https://web.archive.org/web/20070715145029/http:/planetmath.org/encyclopedia/GoldiesTheorem.html https://archive.org/details/topicsringtheory00hers https://archive.org/details/topicsringtheory00hers/page/n10
sameAs	Goldie's theorem Goldie's theorem Goldie's theorem Goldie's theorem
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Cite_book dbt:Cite_journal dbt:Harv dbt:Reflist dbt:Short_description dbt:Abstract-algebra-stub
has abstract	In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (="finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all right ideals. This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain. A consequence of Goldie's theorem, again due to Goldie, is that every semiprime principal right ideal ring is isomorphic to a finite direct sum of prime principal right ideal rings. Every prime principal right ideal ring is isomorphic to a matrix ring over a right Ore domain. (en)
gold:hypernym	Result
prov:wasDerivedFrom	wikipedia-en:Goldie's_theorem?oldid=1096489996&ns=0
page length (characters) of wiki page	4482 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Goldie's_theorem
is Link from a Wikipage to another Wikipage of	Semiprime ring Alfred Goldie Quotient ring Noetherian ring Ring theory Wolverhampton Grammar School List of theorems Uniform module Goldie ring Goldie theorem
is Wikipage redirect of	Goldie ring Goldie theorem
is foaf:primaryTopic of	wikipedia-en:Goldie's_theorem

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