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;p) (mod p) 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;p)≡0 (mod p) for p≥w+3; if w is odd, S(α,β,γ;p)≡rBp≥w (mod p) for p≥w, here r is an explicit rational number independent of p. 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 ring ■ where T,U,V are rings,M,N are U-T,V-U bimodules,respectively.Using the alternative description of left r-module as quintuple(A,B,C;f,g)with A∈modT,B∈modU and C∈modV,f:M_T A→B∈modU,g:N_U B→C∈modV,we shall characterize uniform,hollow and finitely embedded modules overΓ,respectively.Also the radical as well as the socle of r(A⊕B⊕C)is determined.
