N-prime elements and the primality of x − α in D⟦x⟧

Abstract

Let D be an integral domain and D [ ​ [ x ] ​ ] be the power series ring over D. In this paper, we study the primality of x − α in D [ ​ [ x ] ​ ] , where α ∈ D . For this purpose, we generalize the definition of a prime element as follows. For a positive integer N, a nonzero nonunit α ∈ D is called an N-prime element if for any a , b ∈ D , α N | ab implies α | a or α | b . We prove that if α is an N-prime element for some N, then x − α is a prime element in D [ ​ [ x ] ​ ] . Surprisingly, it is shown that the converse also holds when D is a PID, a valuation domain, or a Dedekind domain. In other words, when D is a PID, a valuation domain, or a Dedekind domain, a necessary and sufficient condition for x − α to be a prime element in D [ ​ [ x ] ​ ] is α is an N-prime element in D for some N. This however does not hold for other types of integral domains such as UFDs or Krull domains. We also investigate the N-prime property in an arbitrary integral domain and give other (equivalent) conditions for an element α in the aforementioned types of integral domains to be an N-prime element.