Principal One-Sided Ideals in Ore Polynomial Rings
T. Y. Lam and André Leroy
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 $\sum_ia_it^i$ ($a_i\in K$) is well known to be a principal left ideal domain, although, in the case when $S(K)\neq K$, $R$ 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 $f\in R$ 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.