Let A be a Banach algebra and X a Banach A-bimodule. A derivation from A into X is a bounded linear map satisfying For each we denote by the derivation for all , called an inner derivation. We denote by the space of all derivation from A into X, and by the space of all inner derivation from A into X. The first cohomology group of A with coefficients in X, denoted by , is the quotient space . This first cohomology group of a Banach algebra gives vast information about the structure of A. Let A be a Banach algebra and X be a Banach A-bimodule. Then the module extention Banach algebra corresponding to A and X is , the -direct sum of A and X with the algebra product defind as follows: