Pure projective tilting modules associated with a special ring and Goresntein properties

Abstract

In this paper, we study pure-projective tilting modules and related classes of rings. We introduce the notion of a pure-tilting hereditary ring, namely, a ring over which every ideal is pure-projective tilting, and investigate its structural properties. We prove that a ring R is a pure-tilting hereditary ring if and only if R is hereditary noetherian over a von Neumann regular ring R. In the commutative case, we show that R is a pure one-tilting hereditary ring precisely when R is hereditary noetherian. Using Kaplansky conjecture, we establish a connection between pure-tilting hereditary rings and the hereditary noetherian property of prime factor rings. In category theory, for the torsion pair consisting of Gen of I and the orthogonal class of I in the category of R-modules, we establish that the associated Happel-Reiten-Smalo heart H sub I is a Grothendieck category. We also examine the characterization of Ext-orthogonal classes determined by pure projective tilting modules. In addition, we show that every Gorenstein pure projective tilting module is Gorenstein flat if and only if every Gorenstein pure projective tilting module is strict T-stationary, where T denotes the class of all finitely presented tilting modules. These results establish new links between tilting theory, hereditary ring conditions, and Gorenstein homological structures.