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 $rgbrightarrow rga$ and its right adjoint $rgaotimes_{rgb}-$ 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 $rgb$‎.