Principal one-sided ideals in Ore polynomial rings

Abstract

For an endomorphism S of a division ring K and an S-derivation D on K, the Ore extension R = K[t, S, D] consisting of left polynomials ∑ i ait i (ai ∈ K) is well known to be a principal left ideal domain, although, in the case when S(K) 6= K, R is not a principal right ideal domain (or even a right Ore domain). Using the theory of evaluation of left polynomials on scalars developed in our earlier papers, we define f ∈ R to be a Wedderburn polynomial if f is the minimal polynomial of some (S, D)-algebraic subset of K. We note that Wedderburn polynomials are special cases of “fully reducible” elements in 2-firs. In this paper, we prove a general theorem on 2-firs which implies (in a very explicit way) that the class of Wedderburn polynomials in R = K[t, S, D] is “symmetric” with respect to the left and right ideal structures of R. This 2-fir approach to R also enables us to develop a theory of left roots of the polynomials in R. This theory bears some resemblance to the theory of right roots studied in our earlier papers, but has a number of surprising new features.