Hong, Chan Yong; Kwak, Tai Keun; Rizvi, S. Tariq Rigid ideals and radicals of Ore extensions. (English) Zbl 1090.16010 Algebra Colloq. 12, No. 3, 399-412 (2005). 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) Cited in 15 Documents 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 Keywords:rigid endomorphisms; prime radical; upper nil radical; Ore extensions; rigid ideals Citations:Zbl 0859.16019 PDFBibTeX XMLCite \textit{C. Y. Hong} et al., Algebra Colloq. 12, No. 3, 399--412 (2005; Zbl 1090.16010) Full Text: DOI References: [1] DOI: 10.1016/S0022-4049(96)00011-4 · Zbl 0870.16006 · doi:10.1016/S0022-4049(96)00011-4 [2] DOI: 10.1016/S0022-4049(00)00055-4 · Zbl 0987.16018 · doi:10.1016/S0022-4049(00)00055-4 [3] Ferrero M., Math. J. Okayama Univ. 29 pp 119– [4] Hirano Y., Math. J. Okayama Univ. 20 pp 141– [5] Hirano Y., Publ. Math. Debrecen 54 pp 489– [6] DOI: 10.1016/S0022-4049(99)00020-1 · Zbl 0982.16021 · doi:10.1016/S0022-4049(99)00020-1 [7] DOI: 10.1081/AGB-120016752 · Zbl 1042.16014 · doi:10.1081/AGB-120016752 [8] DOI: 10.1080/00927870008827127 · Zbl 0981.16001 · doi:10.1080/00927870008827127 [9] Huh C., Bull. Korean Math. Soc. 38 pp 623– [10] Kim N. K., Math. Japonica 50 pp 415– [11] Krempa J., Algebra Colloq. 3 pp 289– [12] DOI: 10.1080/00927878108822678 · Zbl 0468.16024 · doi:10.1080/00927878108822678 [13] DOI: 10.1080/00927877708822194 · Zbl 0355.16020 · doi:10.1080/00927877708822194 [14] Rowen L. H., Pure and Applied Math. Series 127, in: Ring Theory (1988) [15] DOI: 10.1090/S0002-9947-1973-0338058-9 · doi:10.1090/S0002-9947-1973-0338058-9 [16] DOI: 10.1016/0022-4049(91)90060-F · Zbl 0747.16001 · doi:10.1016/0022-4049(91)90060-F This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.