arXiv subjects
Publications and source records attributed to Johan Richter.
We study centralizers in certain algebras with valuation in order to generalize results by Hellström and Silvestrov on centralizers in graded algebras. We prove that the centralizer of an element in the studied algebras is a free module over a certain ring. Under further assumptions we obtain that the centralizer is also commutative.
We begin by reviewing a classical result on the algebraic dependence of commuting elements in Weyl algebras. We proceed by describing generalizations of this result to various classes of Ore extensions, both results that have already been published and a new result.
The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id,δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x;σ,δ] is a maximal commutative subring containing R and, in the case when σ=id, we show that it intersects every non-zero ideal of R[x;id,δ] non-trivially. Using this we show that if R is δ-simple and maximal commutative in R[x;id,δ], then R[x;id,δ] is simple. We also show that under some conditions on R the converse holds.