Date of Award
Doctor of Philosophy (PhD)
Professor B. J. Mueller
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.
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.
McDowell, Kenneth Paul, "Commutative Coherent Rings" (1974). Open Access Dissertations and Theses. Paper 1008.