Bhat, V. K. A note on completely prime ideals of Ore extensions. (English) Zbl 1194.16020 Int. J. Algebra Comput. 20, No. 3, 457-463 (2010). Summary: The study of prime ideals has been an area of active research. In recent past a considerable work has been done in this direction. Associated prime ideals and minimal prime ideals of certain types of Ore extensions have been characterized. In this paper a relation between completely prime ideals of a ring \(R\) and those of \(R[x;\sigma,\delta]\) is given; \(\sigma\) is an automorphisms of \(R\) and \(\delta\) is a \(\sigma\)-derivation of \(R\). It is proved that if \(P\) is a completely prime ideal of \(R\) such that \(\sigma(P)=P\) and \(\delta(P)\subseteq P\), then \(P[x;\sigma,\delta]\) is a completely prime ideal of \(R[x;\sigma,\delta]\). It is also proved that this type of relation does not hold for strongly prime ideals. Cited in 8 Documents MSC: 16S36 Ordinary and skew polynomial rings and semigroup rings 16D25 Ideals in associative algebras 16P40 Noetherian rings and modules (associative rings and algebras) Keywords:Ore extensions; automorphisms; derivations; completely prime ideals; strongly prime ideals PDFBibTeX XMLCite \textit{V. K. Bhat}, Int. J. Algebra Comput. 20, No. 3, 457--463 (2010; Zbl 1194.16020) Full Text: DOI References: [1] Anderson D. F., Houston J. Math. 5 pp 451– [2] Anderson D. F., Houston J. Math. 9 pp 325– [3] DOI: 10.1142/S0219498804000782 · Zbl 1060.16029 · doi:10.1142/S0219498804000782 [4] A. Badawi, D. F. Anderson and D. E. Dobbs, Pseudo-valuation Rings, Lecture Notes Pure Appl. Math. 185 (Marcel Dekker, New York, 1997) pp. 57–67. [5] DOI: 10.1080/00927879508825469 · Zbl 0843.13007 · doi:10.1080/00927879508825469 [6] DOI: 10.1080/00927870601141951 · Zbl 1113.13001 · doi:10.1080/00927870601141951 [7] Bhat V. K., Beitrge Algebra Geom. 49 pp 277– [8] DOI: 10.1090/S0002-9939-1990-1025276-5 · doi:10.1090/S0002-9939-1990-1025276-5 [9] DOI: 10.1080/00927870008827069 · Zbl 0959.13001 · doi:10.1080/00927870008827069 [10] DOI: 10.1007/BFb0058802 · doi:10.1007/BFb0058802 [11] Goodearl K. R., An Introduction to Non-Commutative Noetherian Rings (1989) · Zbl 0679.16001 [12] Hedstrom J. R., Pacific J. Math. 4 pp 551– [13] DOI: 10.1081/AGB-120037414 · Zbl 1086.16016 · doi:10.1081/AGB-120037414 [14] DOI: 10.1090/gsm/030 · doi:10.1090/gsm/030 [15] DOI: 10.1016/j.jalgebra.2005.01.014 · Zbl 1072.16024 · doi:10.1016/j.jalgebra.2005.01.014 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.