Much mathematics has been inspired by Cartan's characterization of finite dimensional semisimple algebras over the complex field, as exactly the direct sums of n by n matrix algebras M_n (generalized later to other fields, and usually called the Wedderburn-Artin theorem). We present an infinite dimensional variant of this theorem, valid for algebras of operators on a Hilbert space. We present this in the framework of our ongoing study of one-sided ideals in operator algebras, giving many related results. This is joint work with Almus and Sharma.