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Rigid ideals and radicals of Ore extensions. (English) Zbl 1090.16010

By J. Krempa [Algebra Colloq. 3, No. 4, 289-300 (1998; Zbl 0859.16019)], an endomorphism \(\sigma\) of a ring \(R\) is called rigid if \(a\sigma(a)=0\) implies \(a=0\) for each \(a\in R\). In this paper, a \(\sigma\)-ideal \(I\) of a ring \(R\) is called \(\sigma\)-rigid ideal if \(a\sigma(a)\in I\) implies \(a\in I\) for every \(a\in R\). For an endomorphism \(\sigma\) of a ring, \(\sigma\)-rigid ideals are characterized and related properties are studied. Also connections of the prime radical and the upper nil radical of a ring \(R\) with the prime radical and upper nil radical of the Ore extension \(R[x;\sigma,\delta]\), respectively, are investigated.
Reviewer: J. K. Park (Pusan)

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16W20 Automorphisms and endomorphisms
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16N60 Prime and semiprime associative rings
16D25 Ideals in associative algebras

Citations:

Zbl 0859.16019
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References:

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