## 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 $$p> 0$$.

### MSC:

 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)

### Citations:

Zbl 0518.16003; Zbl 0647.16004; Zbl 1371.16018
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