Ideal Generated by The Coefficient of a Polynomial Over ℤ k , k > 1

Abstract

Let Zk , k > 1, k ∈ N be a commutative ring with unity, polynomial f = a0 + a1x + ⋯ + anx n ∈ Zk[x], ai ∈ Zk. We can construct c(f)= 〈a0, ... , an〉 be an ideal of Zk generated by a0, ... , an . If (a0, ... , an) = 1 or ai unit of Zk for i = 0, ... , n, then c(f) = Zk, for k composite.. For k is prime, because all of the elements in Zk is unit, then c(f) = Zk, for every f ∈ Zk[x].