Let n = 3 k + 2 for some k ∈ N . We investigate the generalized Hyers-Ulam stability of n-homomorphisms and n-derivations on fuzzy ternary Banach algebras related to the generalized Cauchy-Jensen additive functional equation. MSC:39B52, 46S40, 26E50.

LetR=⊕n≥0Rnbe a standard homogeneous Noetherian ring with local base ring(R0,m0)and letMbe a finitely generated gradedR-module. LetHR+i(M)be theith local cohomology module ofMwith respect toR+=⊕n>0Rn. LetSbe a Serre subcategory of the category ofR-modules and letibe a nonnegative integer. In this paper, ifdim⁡R0≤1,then we investigate some conditions under which theR-modulesR0/m0 ⊗R0 HR+i(M),Γm0R(HR+i(M))andHm0R1(HR+i(M))are inSfor alli≥0. Also, we prove that ifdim⁡R0≤2, then the gradedR-moduleHm01(HR+i(M))is inSfor alli≥0. Finally, we prove that ifnis the biggest integer such thatHai(M)∉S, thenHR+i(M)/m0HR+i(M)∈Sfor alli≥n.

In this paper, we study on a special case of generalized matrix algebra that we call it square algebra. According to that Hochschild cohomology play a significant role in Geometry for example in orbifolds, we study the first Hochschild cohomology of the square algebra the vanishing of its.