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...

Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | Torm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This re...

In this note we establish a (non-trivial) lower bound on the degree two entry h2 of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is for Gorenstein h-vectors of the form h = (1, r, h2, r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h2 may assume. In fact, ...

Contravariantly finite resolving subcategories of the category of finitely generated modules have been playing an important role in the representation theory of algebras. In this paper we study contravariantly finite resolving subcategories over commutative rings. The main purpose of this paper is to classify contravariantly finite resolving subcategories over a henselian Gorenstein local ring;...

Moslehian  and Mirmostafaee, investigated the fuzzystability problems for the Cauchy additive functional equation, the Jensen additivefunctional equation and the cubic functional equation in fuzzyBanach spaces. In this paper, we investigate thegeneralized Hyers–-Ulam--Rassias stability of the generalizedadditive functional equation with $n$--variables, in fuzzy Banachspaces. Also, we will show ...