Self-orthogonal modules over coherent rings

Let R be a left coherent ring, S any ring and RωS an (R,S)-bimodule. Suppose ωS has an ultimately closed FP-injective resolution and RωS satisfies the conditions: (1) ωS is finitely presented; (2) The natural map R→ End(ωS) is an isomorphism; (3) ExtSi(ω,ω)=0 for any i≥1. Then a finitely presented left R-module A satisfying ExtRi(A,ω)=0 for any i≥1 implies that A is ω-reflexive. Let R be a left coherent ring, S a right coherent ring and RωS a faithfully balanced self-orthogonal bimodule and n≥0. Then the FP-injective dimension of RωS is equal to or less than n as both left R-module and right S-module if and only if every finitely presented left R-module and every finitely presented right S-module have finite generalized Gorenstein dimension at most n.