International Mathematical Forum, 2, 2007, no. 16, 765 - 770 Ideal Amenability of Second Duals of Banach Algebras M. Eshaghi Gordji (1) , F. Habibian (2) and B. Hayati (3) (1) Department of Mathematics, Faculty of Sciences, Semnan University, Semnan, Iran (2) Department of Mathematics, Isfahan University, Isfahan, Iran (3) Department of Mathematics, Shahid Beheshti University, Tehran, Iran eimail:madjideg@walla.com & fhabibian@ui.ac.ir & bahmanhayati@yahoo.com Abstract In this paper we study the ideal amenability of second duals of Ba- nach algebras. We investigate relations between ideal amenability of the second dual of a Banach algebra with the first and the second Arens products. Mathematics Subject Classification: 46H25 Keywords: Ideally amenable, Banach algebra, Derivation 1 Introduction Let A be a Banach algebra and let A ** be the second dual algebra of A endowed with the first or the second Arens product. Throughout this paper, the first and the second Arens product are respectively denoted by  and ♦. These products can be defined by F G = w * lim i lim j ˆ f i ˆ g j and F ♦G = w * lim j lim i ˆ f i ˆ g j where (f i ) and (g j ) are nets of elements of A such that f i -→ F and g j -→ G in w * -topology (see [A] and [D-H]). If X is a Banach A-bimodule, then a derivation from A into X is a linear map D such that for every a, b ∈A, D(ab)= D(a).b + a.D(b). If x ∈ X and we define δ x : A -→ X by δ x (a)=