An algorithm for the divisors of monic polynomials over a commutative ring

Gilmer and Heinzer proved that given a reduced ring R, a polynomial f divides a monic polynomial in RX] if and only if there exists a direct sum decomposition of R = R₀ ⊕ … ⊕ R_m (m ≤ deg f), associated to a fundamental system of idempotents e₀, … , e_m, such that the component of f in each R_iX] has degree coefficient which is a unit of R_i. We propose to give an algorithm to explicitly find such a decomposition. Moreover, we extend this result to divisors of doubly monic Laurent polynomials.