Hereditary orders in the quotient ring of a skew polynomial ring

Abstract:

Let $K$ be a field, and let $\sigma$ be an automorphism of $K$ of finite order. Let $K(X;\sigma )$ be the quotient ring of the skew polynomial ring $K[X;\sigma ]$. Then any order in $K(X;\sigma )$ which contains $K$ and its center is a valuation ring of the center of $K(X;\sigma )$ is a crossed-product algebra $A_f$, where $f$ is some normalized 2-cocycle. Associated to $f$ is a subgroup $H$ of $\langle \sigma \rangle$ and a graph. In this paper, we determine the connections between hereditary-ness and maximal order properties of $A_f$ and the properties of $H$, $f$ and the graph of $f$.