Let the soluble group \(G=AB\) with finite abelian section rank be the product of two subgroups \(A\) and \(B\). Simple examples show that even if \(G\) is finite, the Fitting subgroup \(F\) of \(G\) need not be factorized, i.e. \(F=AF\cap BF\). In this paper it is proved that nevertheless the Hirsch-Plotkin radical of \(G\) satisfies \(R=A_ 0R\cap B_ 0R\), where \(A_ 0\) and \(B_ 0\) are the Hirsch-Plotkin radicals of \(A\) and \(B\) respectively; and if in addition the set of primes \(p\) for which there is an element of order \(p\) in \(G\) is finite, then the Fitting subgroup \(F\) of \(G\) satisfies \(F=A_ 1F\cap B_ 1F\), where \(A_ 1\) and \(B_ 1\) are the Fitting subgroups of \(A\) and \(B\) respectively. This generalizes a well-known result of E. Pennington and the reviewer that the Fitting subgroup of a finite product of two nilpotent subgroups is factorized and a more general result of the authors and the reviewer that the Hirsch- Plotkin radical of a soluble group with finite abelian section rank, which is the product of two locally nilpotent subgroups, is likewise factorized. – For the proofs of the above results first the finite case is considered and then this is extended to general soluble groups with finite abelian section rank. Here splitting and conjugacy theorems play a role.