Gorenstein Proiective Modules Over Domains

This thesis is concerned with the Gorenstein projective (G-projective for short) modules over domains. Some properties of G-projective modules and some examples of G-projective modules which are not projective are given in the first Chapter. The concept of w-modules over domains was introduced in [38]. It has been generalized to the concept of w-modules over commutative rings in [40]. It is pointed out in this Chapter that G-projective modules are w-modules and, as a counter-example, a w-module which is not G-projcctive has also been given. It is also proved in this Chapter that, if (R. m) is a coherent local ring such that dim(R)=0and M is a finitely presented R-module such that Gpd(M)<∞, then M is in fact G-projective.The concept of G-hereditary rings and G-Dedekind domain was introduced in [32]. A sufficient and necessary condition for a one-dimensional Noetherian domain to be a G-Dedekind domain is given in the second Chapter. Also in this Chapter, without assuming that G-gldim(R)<∞, it is proved that a ring R is a G-hereditary ring if and only if every ideal of R is G-projective. It is also proved in this Chapter that R is a G-Dedekind domain if and only if every nontrivial factor ring of R over any principal ideal is a QF-ring, and furthermore, R is a Dedekind domain if and only if all nontrivial factor rings of R are QF-rings.In the third Chapter, not only an example showing that G-Dedekind domains are not necessarily Noetherian Warfield domains, but also a definition for a special kind of domains-2-DVR is given. A Noetherian local domain is called a2-DVR if its maximal ideal is strongly G-projective. It is proved that a Noetherian domain R is a Warfield domain if and only if. for any maximal ideal M of R, RM is a2-DVR.It is dicussed in the fourth Chapter that under what conditions G-projective modules are projective. It is the Gorenstein homological property of coherent domains whose sclf-FP-injective dimensions are1that is also studied in this Chapter. Furthermore, providing the condition of being integrally closed, it is proved that any finitely generated G-projective module over such rings is projec-tive. Another step further, a characterization of Prufer domains is obtained:a domain R is a Prufer domain if and only if it is coherent, integrally closed and FP-idR(R)≤1.