  For any ring R we construct two triangulated categories, each admitting a
functor from R-modules that sends projective and injective modules to 0. When R
is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree
with each other and with the usual stable module category. Our stable module
categories are homotopy categories of Quillen model structures on the category
of R-modules. These model categories involve generalizations of Gorenstein
projective and injective modules that we derive by replacing finitely presented
modules by modules of type FP-infinity. Along the way, we extend the perfect
duality between injective left modules and flat right modules that holds over
Noetherian rings to general rings by considering weaker notions of injectivity
and flatness.
