If R is the ring of integers in the algebraic number field K and f is a polynomial over R, then f is said to be a permutation polynomial modulo an ideal I (p. p. mod I) of R if the mapping induced on the residue class ring R/I is bijective. Let \(S_ 1(f)\) be the set of nonzero prime ideals P such that f is a p. p. mod P but not mod \(P^ 2\), and let \(S_ 2(f)\) be the set of nonzero prime ideals P such that f is a p. p. mod \(P^ 2\). The main result of the paper gives a characterization of the sets \(S_ 1, S_ 2\) for which there exists an \(f\in R[x]\) with \(S_ i=S_ i(f)\) for \(i=1, 2.\) The proof makes use of the well-known theorem of \textit{M. Fried} [Mich. Math. J. 17, 41-55 (1970; Zbl 0169.377)] that resolved Schur's conjecture. It is also pointed out that the statement of this theorem in Fried's paper is not entirely correct and that the formulation should read as follows: if \(f\in R[x]\) is a p. p. mod P for infinitely many prime ideals P, then f is a composition of polynomials \(ax^ m+b\) with a,b\(\in K\) and Dickson polynomials \(D_ n(c,x)\) with \(c\in R\), \(c\neq 0\).