  We show that the category of projective modules over a graded commutative
ring admits a triangulation with respect to module suspension if and only if
the ring is a finite product of graded fields and exterior algebras on one
generator over a graded field (with a unit in the appropriate degree). We also
classify the ungraded commutative rings for which the category of projective
modules admits a triangulation with respect to the identity suspension.
Applications to two analogues of the generating hypothesis in algebraic
topology are given, and we translate our results into the setting of modules
over a symmetric ring spectrum or S-algebra.
