Some properties of ‎FP-injective‎‎‎ modules over group rings

‎FP-injective modules‎, ‎which are also called absolutely pure modules‎, ‎play an important role in characterizing some classical rings such as semihereditary‎, ‎Noetherian‎, ‎von-Neumann regular‎, ‎and coherent rings‎. ‎These modules have excellent properties over coherent rings similar to injective modules over Noetherian rings‎. ‎In the present article‎, ‎we study this class of modules over the group ring \rga of a group \ga‎, ‎concerning a commutative ring R‎. ‎We show that if \gb is a finite index subgroup of \ga‎, ‎then the restriction of scalars along the natural ring homomorphism \rgb\rightarrow \rga and its right adjoint \rga\otimes_{\rgb}- preserve FP-injective modules‎. ‎We will also examine the properties of FP-injective modules over the group ring of \LHF-groups‎. ‎Next‎, ‎we will switch to the so-called Ding-Chen rings‎. ‎These rings are coherent versions of Iwanaga-Gorenstein rings‎, ‎where Noetherian and self-injectivity are replaced by coherence and self-FP-injectivity‎, ‎respectively‎. ‎In particular‎, ‎we have investigated the ascent and descent of the Ding-Chen property between the rings \rga and \rgb‎.

کلیدواژه ها:

Group ring ، FP-injective module ، Ding-Chen ring