Subalgebras of matrix algebras generated by companion matrices

Let f,g∈Z[X] be monic polynomials of degree n and let C,D∈Mn(Z) be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra Z〈C,D〉 to be a sublattice of finite index in the full integral lattice Mn(Z), in which case we compute the exact value of this index in terms of the resultant of f and g. If R is a commutative ring with identity we determine when R〈C,D〉=Mn(R), in which case a presentation for Mn(R) in terms of C and D is given.