Integrable ODEs on associative algebras

In this paper we give definitions of basic concepts such as symmetries, fir st integrals, Hamiltonian and recursion operators suitable for ordinary dif ferential equations on associative algebras, and in particular for matrix d ifferential equations. We choose existence of hierarchies of first integral s and/or symmetries as a criterion for integrability and justify it by exam ples. Using our componentless approach we have solved a number of classific ation problems for integrable equations on free associative algebras. Also, in the simplest case, we have listed all possible Hamiltonian operators of low order.