Amitsur’s property for skew polynomials of derivation type. (English) Zbl 1383.16028

Various radicals of skew polynomial rings \(R[x,\delta]\) of derivation type are considered. The emphasis in on the \(\delta\)-Amitsur property, which is defined for a radical \(F\) by the condition: for every ring \(R\) and a derivation \(\delta\) of \(R\) we have \(F(R[x,\delta]) = (F(R[x,\delta])\cap R) [x,\delta]\). This is an analogue of a fundamental property established by Amitsur for classical radicals of ordinary polynomial rings. A natural strategy is then to seek an internal description of the ideal \(F(R[x,\delta])\cap R\) of \(R\). By the left \(T\)-nilpotent radideal of a ring \(R\), the authors mean \(T_{l}(R) := \{a\in R : aR \text{ is left } T\text{-nilpotent}\}\). It is shown that \(T_{l}(R[x,\delta]) = (T_{l}(R[x,\delta])\cap R) [x,\delta]\). As a consequence, it is shown that the prime radical has the \(\delta\)-Amitsur property; a result originally obtained by M. Ferrero et al. [J. Lond. Math. Soc., II. Ser. 28, 8–16 (1983; Zbl 0518.16003)]. A radical \(F\) is said to respect finite cyclotomic extensions if the following occurs: for all rings \(A\), and all integer primes \(q\), \(F(A) = F(A[\xi_q])\cap A\); where \(\xi_q \in\mathbb C\) is a primitive \(q\)-th root of unity and \(A[\xi_q] = A\otimes_{\mathbb Z} {\mathbb Z}[\xi_q]\). It is proved that if \(F\) is a radical which respects finite cyclotomic extensions, then \(F\) has the \(\delta\)-Amitsur property. Related results and certain consequences are also discussed. In particular, recovering the result of M. Ferrero asserting that the Jacobson, Levitzki and Brown-McCoy radicals all have the \(\delta\)-Amitsur property [Math. J. Okayama Univ. 29, 119–126 (1987; Zbl 0647.16004)].
Reviewer’s note: A. Smoktunowicz has recently shown in [Isr. J. Math. 219, No. 2, 555–608 (2017; Zbl 1371.16018)] that there exists a Jacobson radical ring ring \(R[x, \delta]\) such that \(R\) is not nil. On the other hand, the Jacobson radical of \(R[x, \delta]\) is of the form \(I[x,\delta]\) for a nil ideal \(I\) of \(R\) provided that \(\delta\) is a locally nilpotent derivation and \(R\) is an algebra over a field of characteristic \(p> 0\).


16S36 Ordinary and skew polynomial rings and semigroup rings
16N60 Prime and semiprime associative rings
16N80 General radicals and associative rings
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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