Lower bounds on the stable range of skew polynomial rings

ABSTRACT

Let R be a ring with an automorphism α and a derivation δ. In this article we provide necessary and sufficient conditions for a skew polynomial ring R[x;α] and differential polynomial ring R[x;δ] to be 2-primal. We compute the Jacobson radical and the set of unit elements of a 2-primal skew polynomial ring R[x;α] and differential polynomial ring R[x;δ]. Also we establish the lower bounds on the stable range of a 2-primal skew polynomial ring R[x;α] and differential polynomial ring R[x;δ]. As an application we show that if R is 2-primal then the nth Weyl algebra over R is 2-primal and in this case $J\left(A_n\right(R\left)\right)=A_n\left(Ni{\ell }_{\ast }\right(R\left)\right)$. As a consequence, we extend and unify several known results of [4], [8], [10], [18], [19], and [22].

KEYWORDS:

• 2-Primal rings
• Jacobson radical
• skew polynomial rings
• stable range

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16S36
• 16N20

Acknowledgment

The author would like to thank the Banach Algebra Center of Excellence for Mathematics, University of Isfahan. Special thanks are due to the referee who read this article very carefully and made many useful suggestions.