Decomposability of monads

Dear categorists, The free ring monad (on, say, Set) may be decomposed into the free monoid monad and the free abelian group monad, connected by a distributive law. Is it known which monads can be decomposed into "simpler" ones (whatever that means), connected by distributive laws? I'm particularly interested in the case of monads describing algebraic theories. Thanks, Miles -- Amazing! You type random strings of letters, and it does what you want! -- Rhiannon Mogridge, on vi