Quotient finite dimensional modules with acc on subdirectly irreducible submodules are noetherian |
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Authors: | Carl Faith |
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Institution: | 199 Longview Ave, Princeton, N.J, 08540 E-mail: cfaithQmath.rutgers.edu |
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Abstract: | A right R-module M is (Goldie) finite dimensional (= f.d.) if M contains no infinite direct sums of submodules.M is quotient f.d. (= q.f.d.) if M/K is f.d. for all submodules K.A submodule I of M is subdirectly irreducible (= SDI) if V is the intersection of all submodules S α of M that properly contain I, then V ≠ I, equivalentlyM/I has simple essential socle V/I. A theorem of Shock 74] states that a q.f.d. right module M is Noether-ian iff every proper submodule of M is contained in a maximal submodule. Camillo 77], proved a companion theorem: M is q.f.d. iff every submodule A ≠ 0 contains a finitely generated (= f.g) submodule S such that A/S has no maximal submodules. Using these two results, and an idea of Camillo 75], we prove the theorem stated in the title. |
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