Jacobson radicals of skew polynomial rings of derivation type. (Q2925379)

Let \(R\) be an associative ring and \(\delta\) a derivation of \(R\), that is, an additive map such that \(\delta(ab)=\delta(a)b+a\delta(b)\) for all \(a,b\in R\). Let \(R[x;\delta]\) denote the skew polynomial ring of derivation type whose elements are the polynomials \(\sum_{i=0}^nr_ix^i\), \(r_i\in R\), where the addition is defined as usual and the multiplication is subject to the relation \(xa=ax+\delta(a)\) for any \(a\in R\). The upper nil radical (that is, the sum of all nil ideals), the Jacobson radical, and the set of all nilpotent elements of \(R\) are denoted by \(\mathrm{Nil}^*(R)\), \(J(R)\), and \(\mathrm{Nil}(R)\), respectively. A ring \(R\) is called strongly \(\pi\)-regular if \(R\) satisfies the descending chain condition on principal right ideals of the form \(aR\supseteq a^2R\supseteq\cdots\) for every \(a\in R\). Rings in which the set of nilpotent elements forms an ideal are called NI rings.NEWLINENEWLINE In this paper the author proves that the following conditions are equivalent for a ring \(R\) and a derivation \(\delta\) of \(R\): (i) \(J(R[x;\delta])\cap R\) is a nil ring; (ii) \(J(R[x;\delta])\cap R\) is a strongly \(\pi\)-regular ring; (iii) \(J(R[x;\delta])\cap\mathrm{Nil}(R)\) is an ideal of \(R\). This allows the author to extend some known results. For example, the author extends Snapper's Theorem and shows that NEWLINE\[NEWLINEJ(R[x;\delta])=\mathrm{Nil}(R[x;\delta])=\mathrm{Nil}(R)[x;\delta]=\mathrm{Nil}^*(R)[x;\delta]=\mathrm{Nil}^*(R[x;\delta])NEWLINE\]NEWLINE if and only if \(\mathrm{Nil}(R)\) is a \(\delta\)-ideal of \(R\) and \(\mathrm{Nil}(R[x;\delta])=\mathrm{Nil}(R)[x;\delta]\). This allows the author to conclude that if \(R[x;\delta]\) is an NI ring, then NEWLINE\[NEWLINEJ(R[x;\delta])=\mathrm{Nil}(R[x;\delta])=\mathrm{Nil}(R)[x;\delta]=\mathrm{Nil}^*(R)[x;\delta]=\mathrm{Nil}^*(R[x;\delta]).NEWLINE\]NEWLINENEWLINENEWLINENEWLINE Reviewer's remark: This is a most enjoyable paper.