Prime factor rings of skew polynomial rings over a commutative dedekind domain

Abstract

This paper is concerned with prime factor rings of a skew polynomial ring over a commutative Dedekind domain. Let P be a non-zero prime ideal of a skew polynomial ring R = D[x; ], where D is a commutative Dedekind domain and is an automorphism of D. If P is not a minimal prime ideal of R, then R/P is a simple Artinian ring. If P is a minimal prime ideal of R, then there are two different types of P , namely, either P = p[x; ] or P = P R, where p is a -prime ideal of D, P is a prime ideal of K[x; ] and K is the quotient field of D. In the first case R/P is a hereditary prime ring and in the second case, it is shown that R/P is a hereditary prime ring if and only if M 2 P for any maximal ideal M of R. We give some examples of minimal prime ideals such that the factor rings are not hereditary or hereditary or Dedekind, respectively.

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