Gorenstein Objects And Their Properties

In this paper, we mainly study Gorenstein objects and their properties in cat-egories of modules and triangulated categories, which consists of two chapters. In Chapter1,we study the cotorsion theory with respect to y-Gorenstein injec-tive and x-Gorenstein projective modules. We prove that if y is a coresolving precovering R-modules class with y-idR(P)<∞for any projective R-module P,then (⊥-(y-gI),y-gI) is a hereditary cotorsion pair, where y-gI is the class of y-Gorenstein injective modules. In Chapter2, we study a special kind of Goren-stein objects and their properties in triangulated categories C, We introduce the notions of strongly n-ξ-Gorenstein projective objects. We also prove that the class of strongly n-ξ-Gorenstein projective objects is closed under finite direct sums and give a simple description of an object with ξ-Gorenstein projective dimension at most n, where ξ is a proper class of triangles.