This paper is devoted to the complete classification of integrable one-component evolution equations whose field variable takes its values in an associative algebra. The proof that the list of non-commutative integrable homogeneous evolution equations is complete relies on the symbolic method. Each equation in the list has infinitely many local symmetries and these can be generated by its recursion (recursive) operator or master symmetry. 1991 Mathematics Subject Classification: 13A50, 16-XX, 22E70, 35A30, 35Q53, 58F07.