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- Goldies_theorem 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.".
- Goldies_theorem wikiPageExternalLink GoldieRing.html.
- Goldies_theorem wikiPageExternalLink GoldiesTheorem.html.
- Goldies_theorem wikiPageID "7738492".
- Goldies_theorem wikiPageLength "4088".
- Goldies_theorem wikiPageOutDegree "27".
- Goldies_theorem wikiPageRevisionID "625692246".
- Goldies_theorem wikiPageWikiLink Alfred_Goldie.
- Goldies_theorem wikiPageWikiLink American_Mathematical_Monthly.
- Goldies_theorem wikiPageWikiLink Annihilator_(ring_theory).
- Goldies_theorem wikiPageWikiLink Artinian_ring.
- Goldies_theorem wikiPageWikiLink Artin–Wedderburn_theorem.
- Goldies_theorem wikiPageWikiLink Ascending_chain_condition.
- Goldies_theorem wikiPageWikiLink Category:Ring_theory.
- Goldies_theorem wikiPageWikiLink Category:Theorems_in_abstract_algebra.
- Goldies_theorem wikiPageWikiLink Classical_ring_of_quotients.
- Goldies_theorem wikiPageWikiLink Essential_extension.
- Goldies_theorem wikiPageWikiLink Essential_submodule.
- Goldies_theorem wikiPageWikiLink Integral_domain.
- Goldies_theorem wikiPageWikiLink Mathematics.
- Goldies_theorem wikiPageWikiLink Matrix_ring.
- Goldies_theorem wikiPageWikiLink Nil_ideal.
- Goldies_theorem wikiPageWikiLink Noetherian_ring.
- Goldies_theorem wikiPageWikiLink Nonsingular_ring.
- Goldies_theorem wikiPageWikiLink Ore_condition.
- Goldies_theorem wikiPageWikiLink Ore_domain.
- Goldies_theorem wikiPageWikiLink Ore_ring.
- Goldies_theorem wikiPageWikiLink Prime_ring.
- Goldies_theorem wikiPageWikiLink Principal_ideal_ring.
- Goldies_theorem wikiPageWikiLink Principal_right_ideal_ring.
- Goldies_theorem wikiPageWikiLink Regular_element.
- Goldies_theorem wikiPageWikiLink Ring_(mathematics).
- Goldies_theorem wikiPageWikiLink Ring_theory.
- Goldies_theorem wikiPageWikiLink Semiprime_ring.
- Goldies_theorem wikiPageWikiLink Semisimple_module.
- Goldies_theorem wikiPageWikiLink Semisimple_ring.
- Goldies_theorem wikiPageWikiLink Singular_submodule.
- Goldies_theorem wikiPageWikiLink Springer-Verlag.
- Goldies_theorem wikiPageWikiLink Springer_Science+Business_Media.
- Goldies_theorem wikiPageWikiLink Uniform_dimension.
- Goldies_theorem wikiPageWikiLink Uniform_module.
- Goldies_theorem wikiPageWikiLinkText "Goldie's theorem".
- Goldies_theorem wikiPageWikiLinkText "right Goldie ring".
- Goldies_theorem hasPhotoCollection Goldies_theorem.
- Goldies_theorem wikiPageUsesTemplate Template:Abstract-algebra-stub.
- Goldies_theorem wikiPageUsesTemplate Template:Citation.
- Goldies_theorem wikiPageUsesTemplate Template:Cite_book.
- Goldies_theorem wikiPageUsesTemplate Template:Cite_journal.
- Goldies_theorem wikiPageUsesTemplate Template:Harv.
- Goldies_theorem wikiPageUsesTemplate Template:Reflist.
- Goldies_theorem subject Category:Ring_theory.
- Goldies_theorem subject Category:Theorems_in_abstract_algebra.
- Goldies_theorem hypernym Result.
- Goldies_theorem 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.".
- Goldies_theorem label "Goldie's theorem".
- Goldies_theorem sameAs m.026bfgw.
- Goldies_theorem sameAs Q5580171.
- Goldies_theorem sameAs Q5580171.
- Goldies_theorem wasDerivedFrom Goldies_theoremoldid=625692246.
- Goldies_theorem isPrimaryTopicOf Goldies_theorem.