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- Hopkins–Levitzki_theorem abstract "In the branch of abstract algebra called ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring R (with 1) is called semiprimary if R/J(R) is semisimple and J(R) is a nilpotent ideal, where J(R) denotes the Jacobson radical. The theorem states that if R is a semiprimary ring and M is an R module, the three module conditions Noetherian, Artinian and \"has a composition series\" are equivalent. Without the semiprimary condition, the only true implication is that if M has a composition series, then M is both Noetherian and Artinian.The theorem takes its current form from a paper by Charles Hopkins and a paper by Jacob Levitzki, both in 1939. For this reason it is often cited as the Hopkins–Levitzki theorem. However Yasuo Akizuki is sometimes included since he proved the result for commutative rings a few years earlier (Lam 2001).Since it is known that right Artinian rings are semiprimary, a direct corollary of the theorem is: a right Artinian ring is also right Noetherian. The analogous statement for left Artinian rings holds as well. This is not true in general for Artinian modules, because there are examples of Artinian modules which are not Noetherian. Another direct corollary is that if R is right Artinian, then R is left Artinian if and only if it is left Noetherian.".
- Hopkins–Levitzki_theorem wikiPageID "29435296".
- Hopkins–Levitzki_theorem wikiPageLength "4262".
- Hopkins–Levitzki_theorem wikiPageOutDegree "27".
- Hopkins–Levitzki_theorem wikiPageRevisionID "701499401".
- Hopkins–Levitzki_theorem wikiPageWikiLink Abstract_algebra.
- Hopkins–Levitzki_theorem wikiPageWikiLink Annihilator_(ring_theory).
- Hopkins–Levitzki_theorem wikiPageWikiLink Artinian_module.
- Hopkins–Levitzki_theorem wikiPageWikiLink Artinian_ring.
- Hopkins–Levitzki_theorem wikiPageWikiLink Ascending_chain_condition.
- Hopkins–Levitzki_theorem wikiPageWikiLink Category:Ring_theory.
- Hopkins–Levitzki_theorem wikiPageWikiLink Category:Theorems_in_abstract_algebra.
- Hopkins–Levitzki_theorem wikiPageWikiLink Commutative_ring.
- Hopkins–Levitzki_theorem wikiPageWikiLink Composition_series.
- Hopkins–Levitzki_theorem wikiPageWikiLink Grothendieck_category.
- Hopkins–Levitzki_theorem wikiPageWikiLink Jacob_Levitzki.
- Hopkins–Levitzki_theorem wikiPageWikiLink Jacobson_radical.
- Hopkins–Levitzki_theorem wikiPageWikiLink Module_(mathematics).
- Hopkins–Levitzki_theorem wikiPageWikiLink Nilpotent_ideal.
- Hopkins–Levitzki_theorem wikiPageWikiLink Noetherian_module.
- Hopkins–Levitzki_theorem wikiPageWikiLink Noetherian_ring.
- Hopkins–Levitzki_theorem wikiPageWikiLink Ring_theory.
- Hopkins–Levitzki_theorem wikiPageWikiLink Semisimple_algebra.
- Hopkins–Levitzki_theorem wikiPageWikiLink Semisimple_module.
- Hopkins–Levitzki_theorem wikiPageWikiLink Tsit_Yuen_Lam.
- Hopkins–Levitzki_theorem wikiPageWikiLink Yasuo_Akizuki.
- Hopkins–Levitzki_theorem wikiPageWikiLinkText "Akizuki-Hopkins-Levitzki Theorem".
- Hopkins–Levitzki_theorem wikiPageWikiLinkText "Akizuki–Hopkins–Levitzki theorem".
- Hopkins–Levitzki_theorem wikiPageWikiLinkText "Hopkins–Levitzki theorem".
- Hopkins–Levitzki_theorem wikiPageWikiLinkText "semiprimary ring".
- Hopkins–Levitzki_theorem wikiPageUsesTemplate Template:Citation.
- Hopkins–Levitzki_theorem wikiPageUsesTemplate Template:Harv.
- Hopkins–Levitzki_theorem wikiPageUsesTemplate Template:Reflist.
- Hopkins–Levitzki_theorem subject Category:Ring_theory.
- Hopkins–Levitzki_theorem subject Category:Theorems_in_abstract_algebra.
- Hopkins–Levitzki_theorem type Redirect.
- Hopkins–Levitzki_theorem type Theorem.
- Hopkins–Levitzki_theorem comment "In the branch of abstract algebra called ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring R (with 1) is called semiprimary if R/J(R) is semisimple and J(R) is a nilpotent ideal, where J(R) denotes the Jacobson radical.".
- Hopkins–Levitzki_theorem label "Hopkins–Levitzki theorem".
- Hopkins–Levitzki_theorem sameAs Q17027556.
- Hopkins–Levitzki_theorem sameAs משפט_הופקינס-לויצקי.
- Hopkins–Levitzki_theorem sameAs ホプキンス・レヴィツキの定理.
- Hopkins–Levitzki_theorem sameAs m.0dscvhh.
- Hopkins–Levitzki_theorem sameAs Q17027556.
- Hopkins–Levitzki_theorem wasDerivedFrom Hopkins–Levitzki_theorem?oldid=701499401.
- Hopkins–Levitzki_theorem isPrimaryTopicOf Hopkins–Levitzki_theorem.