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Jordan, C. R. and Jordan, D. A.
(1976).
DOI: https://doi.org/10.1080/00927877608822125
Abstract
A well known result on polynomial rings states that, for a given ring , if
has no non-zero nil ideals then the polynomial ring
(x) is semiprimitive, see for example (5) p.12. In this note we study Ore extensions of the form
(x,δ), where δ is an automorphism on the ring
, with the aim of relating the question of the semiprimitivity of
(x,δ) to the presence of non-zero nil ideals in
. In particular we show that under certain chain conditions the Jacobson radical of
(x,δ) consists precisely of polynomials over the nilpotent radical of
. Without restriction on
we show that if δ has finite order then
(x,δ) is semiprimitive if
has no nil ideals. Some conditions are required on
and δ for results of the above nature to be true, as illustrated in §5 by an example of a semiprimitive ring
having an automorphism δ of infinite order such that
(x,δ) has nil ideals.