n-Matlis cotorsion modules and n-matlis domains

Abstract

Let [Formula: see text] be a domain with its field [Formula: see text] of quotients, [Formula: see text] an [Formula: see text]-module and [Formula: see text] a fixed non-negative integer. Then [Formula: see text] is called [Formula: see text]-Matlis cotorsion if [Formula: see text] for any integer [Formula: see text]. Also [Formula: see text] is said to be [Formula: see text]-Matlis flat if [Formula: see text] for any [Formula: see text]-Matlis cotorsion [Formula: see text]-module [Formula: see text]. We proved that [Formula: see text] is a complete hereditary cotorsion theory, where [Formula: see text] (respectively, [Formula: see text]) denotes the class of all [Formula: see text]-Matlis flat (respectively, [Formula: see text]-Matlis cotorsion) [Formula: see text]-modules. In this paper, it is proved that [Formula: see text] is an [Formula: see text]-Matlis domain if and only if epic images of [Formula: see text]-Matlis cotorsion [Formula: see text]-modules are again [Formula: see text]-Matlis cotorsion if and only if [Formula: see text]-Matlis flat [Formula: see text]-modules are of projective dimension [Formula: see text] if and only if [Formula: see text] if and only if [Formula: see text].