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

Let D be an integral domain and (Formula presented.) be the power series ring over D. In this paper, we study the primality of (Formula presented.) in (Formula presented.), where (Formula presented.). For this purpose, we generalize the definition of a prime element as follows. For a positive integer N, a nonzero nonunit (Formula presented.) is called an N-prime element if for any (Formula presented.) implies (Formula presented.) or (Formula presented.). We prove that if α is an N-prime element for some N, then (Formula presented.) is a prime element in (Formula presented.). 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 (Formula presented.) to be a prime element in (Formula presented.) 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.