Abstract. Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and n a fixed positive integer.then R is commutative. We also examine the case where R is a semiprime ring.

Abstract. This paper continues a line investigation in [1]. Let A be a K-algebra and M an A/K-bimodule. In [5] Hamaguchi gave a necessary and sufficient condition for gDer(A, M ) to be isomorphic to BDer(A, M ). The main aim of this paper is to establish similar relationships for generalized (σ, τ )-derivations.

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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.