Please use this identifier to cite or link to this item:
http://hdl.handle.net/11375/5660
Title: | Commutative Coherent Rings |
Authors: | McDowell, Paul Kenneth |
Advisor: | Mueller, B. J. |
Department: | Mathematics |
Keywords: | Mathematics;Mathematics |
Publication Date: | May-1974 |
Abstract: | <p>Those modules over a commutative Noetherian ring which are finitely generated (and therefore automatically finitely presented) have especially pleasant properties. For example, any such module has a finitely generated projective resolution. Furthermore, any ideal contained in the set of zero-divisors of a non-zero finitely generated module M is actually annihilated by some non-zero element of M. Now the property that any finitely presented module has a finitely generated projective resolution actually characterizes coherent rings. Those commutative coherent rings whose non-zero finitely presented modules posses the second property mentioned above with respect to finitely generated ideals are herein entitled "pseudo-Noetherian" rings. This thesis is devoted to the study of these rings.</p> <p>It is demonstrated that a faithfully flat directed colimit of such rings is again pseudo-Noetherian and this observation leads to non-trivial examples of pseudo-Noetherian rings. Equipped with a suitable definition for the "depth" of a non-zero finitely presented module M over a local pseudo-Noetherian ring R one may establish the following extensions of results known in the Noetherian situation: Depth M equals the length of any maximal R-sequence on M. Moreover, if M = R, this number equals the supremum of the projective dimensions of those finitely presented modules which have finite projective dimension. Furthermore, if M has finite projective dimension, (p. dim M) + (depth M) = depth R. These last two statements may be sharpened by substituting "Gorenstein dimension" for "projective dimension" wherever the latter occurs.</p> |
URI: | http://hdl.handle.net/11375/5660 |
Identifier: | opendissertations/1008 1590 931168 |
Appears in Collections: | Open Access Dissertations and Theses |
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