In this thesis we investigate a generalization of purity for modules which has applications to the study of direct sums of cyclic modules over valuation domains. Kaplansky {Trans. Amer. Math. Soc. 72 (1952), page 332, footnote 4} suggested the following as a possible generalization for the purity of a submodule A of an R-module B for a general ring R : for any b + A (ELEM) B/A there exists an a (ELEM) A such that Ann(b + A) = Ann(b - a). We call a submodule with this property cyclically-pure in B. The notion is stronger than RD-purity (relative-divisibility), but coincides with RD-purity and purity in the sense of Cohn for Dedekind domains We investigate in the thesis some general properties of cyclic-purity as well as some of the homological aspects associated with it. In particular, we look at the relationship between cyclic-purity and purity, showing that Prufer and noetherian domains can be characterized in terms of this relationship. We also consider the cyclically-pure-projective modules, which are just the summands of direct sums of cyclics (direct sums of cyclics over local domains), and a cyclically-pure-projective dimension of a module, which measures, in a sense, how far a module is from being a direct sum of cyclics. We show that some basic results from the theory of projective dimensions carry over, with appropriate modifications, to cyclically-pure-projective dimensions, and by restricting to valuation domains, we obtain results connecting the cyclically-pure-projective dimension of a module with the cardinality of a set of generators of minimal cardinality. The results are then applied in the investigation of submodules of direct sums of cyclics over valuation domains / acase@tulane.edu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_26825 |
| Date | January 1983 |
| Contributors | Simmons, James Hamilton (Author) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Rights | Access requires a license to the Dissertations and Theses (ProQuest) database., Copyright is in accordance with U.S. Copyright law |
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