Gorenstein AC-Projective and AC-Injective Modules over Formal Triangular Matrix Rings

Abstract

Let [Formula: see text] and [Formula: see text] be rings and [Formula: see text] a [Formula: see text]-bimodule. If [Formula: see text] is flat and [Formula: see text] is finitely generated projective (resp., [Formula: see text] is finitely generated projective and [Formula: see text] is flat), then the characterizations of level modules and Gorenstein AC-projective modules (resp., absolutely clean modules and Gorenstein AC-injective modules) over the formal triangular matrix ring [Formula: see text] are given. As applications, it is proved that every Gorenstein AC-projective left [Formula: see text]-module is projective if and only if each Gorenstein AC-projective left [Formula: see text]-module and [Formula: see text]-module is projective, and every Gorenstein AC-injective left [Formula: see text]-module is injective if and only if each Gorenstein AC-injective left [Formula: see text]-module and [Formula: see text]-module is injective. Moreover, Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring [Formula: see text] are studied.