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Nilradicals of power series rings and nil power series rings. (English) Zbl 1086.16010
Let \(R\) be a ring, not necessarily with an identity, and let \(X\) be a nonempty set of indeterminates. \(R[\![X]\!]\) and \(R\{X\}\) denote the power series over \(R\) with \(X\) commuting and when \(X\) is noncommuting, respectively. Earlier results on the nilradical of these rings [E. R. Puczyłowski and A. Smoktunowicz, Isr. J. Math 110, 317-324 (1999; Zbl 0934.16016)] are extended.
In particular, it is shown that \(N_2(R[\![X]\!])=N_*(R[\![X]\!])=N^*(R[\![X]\!])\) where \(N^*\) denotes the nil radical, \(N_*\) denotes the prime radical and \(N_2(A)\) is the ideal of the ring \(A\) with \(N_2(A)/N_1(B)=N_1(A/N_1(A))\). Here \(N_1(B)\) denotes the sum of the nilpotent ideals of the ring \(B\). If \(R\) is a nil ring of index \(n\geq 2\), then it is shown that \(R[\![X]\!]\) is a nil ring of index \(\leq n!\). It is also shown that if \(R\) is a PI ring of bounded index, then so is \(R[\![X]\!]\).
These are two of the main results. Amongst others, characterizations of when \(R[\![X]\!]\) is nil or when \(N_*(R)[\![X]\!]\) is nil are given. The paper contains a number of examples to show that the converses of many results are not valid.

16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
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