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Noncommutative Prüfer rings and some generalizations

University of British Columbia

Noncommutative Prfifer rings appear naturally when one wants to transfer the known results for rings which arise in algebraic geometry (such as Dedekind, Krull and Priifer, valuation rings ...) to noncommutative rings. We remove the left-right symmetry condition of the noncommutative Prfifer rings introduced by Alajbegovic and Dubrovin, and introduce three natural generalizations, semi-Prfifer rings, right w-semi-Prfifer rings, and right w-Prfifer rings. We study the relations between the four concepts, and present the various properties that characterize them. We formulate and prove the basic facts for those rings (decompositions of such rings; Morita invariants of these notions; relations with some other notions). A new module-theoretic characterization of semiprime right Goldie rings is achieved by using the newly-defined concept of strongly compressible modules. The result is used to provide new characterizations of semiprime Goldie (prime right Goldie, or prime Goldie) rings, and right w-semi-Prfifer (semi-Prfifer, right w-Prfifer,or Prfifer) rings. In particular, the characterization of semiprime Goldierings of Lopez-Permouth, Rizvi, and Yousif using weakly-injective modules is an easy corollary of our results. We also study modules over noncommutative Priifer rings. It is shown that a module over a noncommutative Prfiferring has projective dimension at most one if and only if it is the union of a well-ordered continuous chain of submodules with each factor of the chain a finitely presented cyclic module. The result is used to present a characterization of divisible modules with projective dimension at most one over noncommutative Priifer rings, which generalizes a known result of L.Fuchs.

Thesis/Dissertation

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2008-09-17

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http://hdl.handle.net/2429/2140

Mathematics

University of British Columbia

UBCV

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