Abstract: Let K* be an associative algebra over a field F with identity u , and let u, e 1 , e 2 , … , be a basis for K*. Denote by K the linear space, over F , spanned by the e i ,i = 1, 2, … . Then for x, y in K, xy = αu + a, where a ∈ K. Define h(x, y) = α and x.y = a. With respect to the operation thus defined, K becomes an algebra over F satisfying 1 Further, the bilinear form h(x, y) is associative on K . Any algebra, over a field F , which possesses an associative bilinear form h(x, y) and satisfies (1) will be called a algebra. It is not difficult to show that any algebra K can be obtained from a unique associative algebra K* with identity by the process described above. The algebra K* will be called the associated associative algebra of K .