Commutative Coherent Rings

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>