An. S ¸tiint ¸. Univ. Al. I. Cuza Ia¸ si. Mat. (N.S.) Tomul LXIII, 2017, f. 2 Permanent weak module amenability of semigroup algebras Abasalt Bodaghi · Massoud Amini · Ali Jabbari Received: 29.VI.2013 / Revised: 22.I.2014 / Accepted: 29.I.2014 Abstract We employ the fact that L 1 (G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ 1 (S) is n-weakly module amenable as an ℓ 1 (E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras. Keywords Banach modules · module derivation · n-weak module amenability · inverse semigroup · module projective tensor product Mathematics Subject Classification (2010) 43A20 · 46H25 1 Introduction A Banach algebra A is amenable if H 1 (A,X ∗ )= {0} for every Banach A-module X, where H 1 (A,X ∗ ) is the first Hochschild cohomology group of A with coefficients in X ∗ . The notion is introduced by Johnson in [17]. Dales et al introduced the notion of n- weak amenability of Banach algebras in [12]. A Banach algebra A is n-weakly amenable if H 1 (A, A (n) )= {0}, where A (n) is nth dual space of A (1-weak amenability is called Abasalt Bodaghi Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran E-mail: abasalt.bodaghi@gmail.com Massoud Amini Department of Mathematics, Tarbiat Modares University, Tehran 14115-134, Iran E-mail: mamini@modares.ac.ir Ali Jabbari Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran E-mail: ali.jabbari@iauardabil.ac.ir; jabbari al@yahoo.com 287