In this paper we shall apply modules of generalized fractions to extend the notion of the grade of an ideal, and to obtain characterizations of Gorenstein rings.

Let P and Q be finite partially ordered sets with |P | = |Q| = d, and C(P ) ⊂ R and C(Q) ⊂ R their chain polytopes. The twinned chain polytope of P and Q is the normal Gorenstein Fano polytope Γ(C(P ),−C(Q)) ⊂ R which is the convex hull of C(P )∪(−C(Q)). In this paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain po...