Quasiprojective modules over integral domains

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Title 
Quasiprojective modules over integral domains 
Author 
Alexeev, Dmitri M 
School 
Tulane University 
Academic Field 
Mathematics 
Abstract 
We study quasiprojective modules over integral domains. A module M is called quasiprojective if it has the projective property relative to all exact sequences of the form 0 → N → M → M/N → 0, where N is a submodule of M. Quasiprojective modules have been introduced by Miyashita as a generalization of projective modules The main goals of our research are to generalize the results on quasiprojective modules over valuation domains to arbitrary integral domains and to study special types of quasiprojective modules, e.g. uniserial modules The dissertation consists of two parts The first part is concerned with quasiprojective modules over general domains. The main results of the first part are the following. (1) The socalled 1½generated ideals are quasiprojective, moreover, projective. (2) The quotient field Q of an integral domain R is a quasiprojective Rmodule if and only if every proper submodule of Q is complete in its R topology. (3) Integral domains all of whose ideals are quasiprojective are exactly the almost maximal Prufer domains The second part of the dissertation is primarily devoted to quasiprojective uniserial modules over valuation domains. The main results of the second part are the following. (1) Uniserial module U is quasiprojective if and only if it is weakly quasiprojective and an additional technical requirement is satisfied. (2) For torsion free modules of rank 1, quasiprojectivity is equivalent to the weak quasiprojectivity, and the latter is determined by completeness of certain endomorphism rings in their ring topologies. (3) The archimedean ideals of a valuation domain R with nonprincipal maximal ideal P are quasiprojective if and only if R/K is complete in the R/ Ktopology for each archimedean ideal K, not isomorphic to P In conclusion we investigate the influence of quasiprojectivity on the decomposability of modules over valuation domains as well as on the properties of direct sums of such modules. We show that a torsionfree quasiprojective module M over a valuation domain which has a dense basic submodule is completely decomposable and that direct sums of ℵ generated uniserial modules of cardinality less than ℵ are quasiprojective 
Language 
eng 
Advisor(s) 
Fuchs, Laszlo 
Degree Date 
2000 
Degree 
Ph.D 
Publisher 
Tulane University 
Publication Date 
2000 
Source 
43 p., Dissertation Abstracts International, Volume: 6104, Section: B, page: 1976 
Identifier 
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Rights 
Copyright is in accordance with U.S. Copyright law 
Contact Information 
acase@tulane.edu 
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