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Nil ideals in the near-ring of formal power series. (English) Zbl 1143.13300
Dorfer, G. (ed.) et al., Proceedings of the 73rd workshop on general algebra “73. Arbeitstagung Allgemeine Algebra”, 22nd conference of young algebraists, Alps-Adriatic-University of Klagenfurt, Austria, February 1–4, 2007. Klagenfurt: Verlag Johannes Heyn (ISBN 978-3-7084-0303-8/pbk). Contributions to General Algebra 18, 97-106 (2008).
Summary: First the question is considered, when a formal power series with nilpotent coefficients is nilpotent with respect to substitution \(\circ\). A positive answer can be given in case of principal ideal rings. To get a survey about nil near-ring ideals, connections between the ideal structure of the coefficient ring \(R\) and the near-ring \(N= (R_0[[X]],+,\circ)\) of formal power series of positive order are studied. After determining some radicals \(\gamma\) in \(N\) the relation \(R\in\gamma\Rightarrow R_0[[X]]\in\gamma\) is examined. For principal ideal rings \(R\) one gets the result, that \(N\) is a Jacobson radical near-ring for the various Jacobson radicals if \(R\) is a nil ring.
For the entire collection see [Zbl 1134.08001].
13A15 Ideals and multiplicative ideal theory in commutative rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
13J05 Power series rings