In this paper, the notions of simple, very simple and indecomposable autonomous behaviour are introduced and characterized. By resorting to some recent results about the direct sum decomposition of (linear, time-invariant, differential) behaviours (Bisiacco and Valcher 2001), as well as to the well-known primary decomposition theorem for finitely generated modules (Hartley and Hawkes 1970), it is shown that every autonomous behaviour can be expressed as a direct sum of indecomposable components, which are just cyclic modules of order p(nu), for some irreducible polynomial p and some positive integer nu. The non-uniqueness of this result is also discussed. Finally, this decomposition is interpreted in terms of modal analysis, and related to the results that can be obtained by mean of a state-space realization of the behaviour.