Zhao (2003a) first established a congruence for any odd prime p〉3, S(1,1,1 ;p)=-2Bp-3 (mod p), which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S(α,β, γ,ρ) (modp) is considered for all positive integers α,β, γ. We refer to w=α+β+ γ as the weight of the sum, and show that if w is even, S(α,β, γ,ρ)=0 (mod p) for p≥w+3; if w is odd, S(α,β, γ,ρ)=-rBp-w (mod p) for p≥w, here r is an explicit rational number independent ofp. A congruence of Catalan number is obtained as a special case.
In this paper we carry out a study of modules over a 3 × 3 formal triangular matrix ringГ=(T 0 0 M U 0 N×UM N V)where T, U, V are rings, M, N are U-T, V-U bimodules, respectively. Using the alternative description of left Г-module as quintuple (A, B, C; f, g) with A ∈ mod T, B ∈ mod U and C ∈ mod V, f : M ×T A →B ∈ mod U, g : N ×U B → C ∈ mod V, we shall characterize uniform, hollow and finitely embedded modules over F, respectively. Also the radical as well as the socle of r (A + B + C) is determined.