| Affiliation: | (1) Department of Mathematics, Naruto University of Education, Takashima, Naruto 772-8502, Japan;(2) Institute of Mathematics of the Romanian Academy, PO Box 1-764, Ro-70700 Bucharest, Romania;(3) Department of Mathematics, University of Bucharest, Str. Academiei 14, 10109 Bucharest, Romania |
| Abstract: | Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let Q[X, ] be the skew polynomial ring over Q in an indeterminate X, where is an automorphism of Q. Consider the natural map  from Q[X,]XQ[X,] to Q, where Q[X,]XQ[X,] is the localization of Q[X,] at the maximal ideal XQ[X,] and set  , the complete inverse image of R by . It is shown that  is a Dubrovin valuation ring of Q(X,) (the quotient ring of Q[X,]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism is classified into five types, in order to study the structure of  (the value group of  ). It is shown that there is a commutative valuation ring R with automorphism which belongs to each type and which makes  Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional.Presented by A. VerschorenMathematics Subject Classifications (2000) 16L99, 16S36, 16W60. |
