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Prime radicals of formal power series rings. (English) Zbl 1001.16011
Let $$R[X]$$ and $$R[[X]]$$ denote respectively the ring of polynomials and the ring of formal power series in a set $$X$$ of commuting indeterminates with coefficients in a ring $$R$$. It is shown that if any one of the three rings $$R$$, $$R[X]$$, $$R[[X]]$$ is semi-prime then so also are the other two. The paper then considers the condition that the prime radical of a ring is nilpotent, with a typical result being that this property is a Morita invariant.

##### MSC:
 16N60 Prime and semiprime associative rings 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 16S36 Ordinary and skew polynomial rings and semigroup rings 16D25 Ideals in associative algebras 16N40 Nil and nilpotent radicals, sets, ideals, associative rings