Prime radicals of formal power series rings. (English) Zbl 1001.16011

Huh, Chan; Kim, Hong Kee; Lee, Dong Su; Lee, Yang

Bull. Korean Math. Soc. 38, No. 4, 623-633 (2001).

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.

Reviewer: A.W.Chatters (Bristol)

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

Keywords:

rings of polynomials; rings of formal power series; semi-prime rings; prime radical; nilpotent rings; Morita invariants