1. Introduction. The general study of algebras of operators on Hilbert space has led to the investigation of rings of operators, also called W*algebras or von Neumann algebras. These are self-adjoint, weakly closed algebras of operators which contain the identity. If the center of a ring (center in the algebraic sense) consists only of scalar multiples of the identity, then the ring is a factor. Factors have been studied extensively and divided into types by Murray and von Neumann [5; 6]. In his work on reduction theory [9], von Neumann has considered the decomposition of a ring with respect to various subalgebras contained in its center. When the subalgebra actually is the center, then the rings making up the decomposition are factors. The question of decomposition with respect to a subalgebra which is not the center of the ring, but is maximal abelian in its commutant, is also of interest. Here, each of the rings in the decomposition is isomorphic to the ring of all bounded operators on some Hilbert space [4]. This sort of decomposition is not unique, but rather depends essentially upon the choice of the maximal abelian subalgebra. However, not much is known about these subalgebras, even in the case of a continuous factor of finite type, or a type II1 factor. In this paper we restrict ourselves to the study of approximately finite l1b factors, that is, those which are generated by a sequence of factors