A note on the semiprimitivity of Ore extensions

Jordan, C. R. and Jordan, D. A. (1976). A note on the semiprimitivity of Ore extensions. Communications in Algebra, 4(7) pp. 647–656.

DOI: https://doi.org/10.1080/00927877608822125

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.

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