In this paper, the relationship between the decomposition of (linear, time-invariant, differential) behaviors and the solvability of certain two-sided diophantine equations is explored. The possibility of expressing a behavior as the sum of two sub-behaviors, endowed with a finite dimensional (and hence autonomous) intersection, one of which is a priori chosen, proves to be related to the solvability of a particular two-sided diophantine equation. In particular, the existence of a direct sum decomposition is equivalent to the solvability of a two-sided Bezout equation, and hence to the internal skew-primeness of a suitable matrix pair.