A note on semiprimitivity of ore extensions

Abstract

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