  We introduce what is meant by an AC-Gorenstein ring. It is a generalized
notion of Gorenstein ring which is compatible with the Gorenstein AC-injective
and Gorenstein AC-projective modules of Bravo-Gillespie-Hovey. It is also
compatible with the notion of $n$-coherent rings introduced by Bravo-Perez: So
a $0$-coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the
sense of Iwanaga, while a $1$-coherent AC-Gorenstein ring is precisely a
Ding-Chen ring. We show that any AC-Gorenstein ring admits a stable module
category that is compactly generated and is the homotopy category of two
Quillen equivalent abelian model category structures. One is projective with
cofibrant objects the Gorenstein AC-projective modules while the other is an
injective model structure with fibrant objects the Gorenstein AC-injectives.
