Let M be a Γ-ring and S ⊆ M. A mapping f : M → M is called strong commutativity preserving on S if [f (x), f (y)]α = [x, y]α, for all x, y ∈ S, α ∈ Γ. In the present paper, we investigate the commutativity of the prime Γ-ring M of characteristic not 2 with center Z(M) = (0) admitting a derivation which is strong commutativity preserving on a nonzero square closed Lie ideal U of M. Moreover, we also obtain a related result when a mapping d is assumed to be a derivation on U satisfying the condition d(u) •α d(v) = u •α v, for all u, v ∈ U , α ∈ Γ.

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In this paper, we generalize the notion of n-weak module amenability of a Banach algebra A which is a Banach module over another Banach algebra U with compatible actions to that of (σ ) − n-weak module amenability for n ∈ N and σ ∈ Hom U (A). We also investigate the relation between this new concept of amenability of A and the quotient Banach algebra A/J where J is the closed ideal of A generated by elements of the form (a · α)b − a(α · b) for a, b ∈ A and α ∈ U . As a consequence, we show that the semigroup algebra l 1 (S) is (σ )-(2n + 1)-weakly module amenable as an l 1 (E)-module for each n ∈ N and σ ∈ Hom l 1 (E) (l 1 (S)), where S is an inverse semigroup with the set of idempotents E.

Abstract. The Hyers-Ulam-Rassias stability of the k-th partial ternary quadratic derivations is investigated in non-Archimedean Banach ternary algebras and non-Archimedean C * −ternary algebras by using the fixed point theorem.

The notion of module amenability for a class of Banach algebras, which could be considered as a generalization of Johnson’s amenability, was introduced by Amini in [Module amenability for semigroup algebras, Semigroup Forum 69 (2004) 243–254]. The weak module amenability of the triangular Banach algebra [Formula: see text], where [Formula: see text] and [Formula: see text] are Banach algebras (with [Formula: see text]-module structure) and [Formula: see text] is a Banach [Formula: see text]-module, is studied by Pourabbas and Nasrabadi in [Weak module amenability of triangular Banach algebras, Math. Slovaca 61(6) (2011) 949–958], and they showed that the weak module amenability of [Formula: see text] triangular Banach algebra [Formula: see text] (as an [Formula: see text]-bimodule) is equivalent with the weak module amenability of the corner algebras [Formula: see text] and [Formula: see text] (as Banach [Formula: see text]-bimodules). The main aim of this paper is to investigate the module [Formula: see text]-amenability and weak module [Formula: see text]-amenability of the triangular Banach algebra [Formula: see text] of order three, where [Formula: see text] and [Formula: see text] are [Formula: see text]-module morphisms on [Formula: see text]. Also, we give some results for semigroup algebras.