N-prime elements and the primality of x-α in in D[[x]]

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- a inD[[x]], wherea. D. For this purpose, we generalize the definition of a prime element as follows. For a positive integer N, a nonzero nonunit a. D is called an N-prime element if for any a, b. D, aN | ab implies a | a or a | b. We prove that if a is an N-prime element for some N, then x - a 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 - a to be a prime element in D[[x]] is a 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 a in the aforementioned types of integral domains to be an N-prime element.