Non-Unique Factorization of Polynomials Over Residue Class Rings of the Integers

Abstract

We investigate non-unique factorization of polynomials in ℤ p n [x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, ℤ p n [x] is atomic. We reduce the question of factoring arbitrary nonzero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of ℤ p n [x] is a direct sum of monoids corresponding to irreducible polynomials in ℤ p [x], and we show that each of these monoids has infinite elasticity. Moreover, for every m ∈ ℕ, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.