×

Prime ideals in near-rings. (English) Zbl 0796.16035

Authors’ abstract: An ideal \(I\) of a near-ring \(R\) is a type one prime ideal if whenever \(aRb\subseteq I\), then \(a\in I\) or \(b\in I\). This paper considers the connections between prime ideals and type one prime ideals in near-rings. It also develops properties of type one prime ideals, gives several examples illustrating where prime and type one prime are not equivalent, and investigates the properties of the type one prime radical. Several different types of conditions are given which guarantee that a prime ideal is type one. The class of all near-rings for which each prime ideal is type one is investigated and many examples of such near-rings are exhibited. Various localized distributivity conditions are found which are useful in establishing when prime ideals will be type one prime.

MSC:

16Y30 Near-rings
16D25 Ideals in associative algebras
16N80 General radicals and associative rings
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. André, Noncommutative geometry, near-rings and near-fields, in: Near-Rings and Near-Fields (ed. by G. Betsch), Proceedings, Tübingen 1985. (Math. Studies, vol. 137, pp. 1–13) Amsterdam, New York, Oxford, Tokyo, North-Holland, 1987.
[2] J. Beidleman, Strictly prime distributively generated near-rings, Math. Z. 100 (1967), 97–105. · Zbl 0154.27101 · doi:10.1007/BF01110786
[3] H. Bell and G. Mason, On derivations in near-rings, in: Near-Rings and Near-Fields (ed. by G. Betsch), Proceedings, Tübingen 1985. (Math. Studies, vol. 137, pp. 31–35) Amsterdam, New York, Oxford, Tokyo, North-Holland, 1987. · Zbl 0619.16024
[4] G. Birkenmeier and H. Heatherly, Medial near-rings, Mh. Math. 107 (1989), 89–110. · Zbl 0685.16021 · doi:10.1007/BF01300916
[5] G. Birkenmeier and H. Heatherly, Polynomial identity properties for near-rings on certain groups, Near-ring Newsletter 12 (1989), 5–15. · Zbl 0685.16021
[6] G. Birkenmeier and H. Heatherly, Left self distributive near-rings, J. Austral. Math. Soc. 49 (1990), 273–296. · Zbl 0713.16023 · doi:10.1017/S144678870003055X
[7] G. Birkenmeier and H. Heatherly, Permutation identity near-rings and ”localized” distributivity conditions, Mh. Math. 111 (1991), 265–285. · Zbl 0736.16023 · doi:10.1007/BF01471181
[8] G. Booth, N. Groenewald, and S. Veldsman, A Kurosh-Amitsur radical for near-rings, Comm. Alg. 18 (1990), 3111–3122. · Zbl 0706.16025 · doi:10.1080/00927879008824063
[9] J. Clay, The near-rings on groups of low order, Math. Z. 104 (1968), 364–371. · Zbl 0153.35704 · doi:10.1007/BF01110428
[10] R. Faudree, Groups in which each element commutes with its endomorphic images, Proc. Amer. Math. Soc. 27 (1971), 236–240. · Zbl 0209.33202 · doi:10.1090/S0002-9939-1971-0269737-3
[11] Y. Fong and J.D.P. Meldrum,The endomorphism near-rings of the symmetric groups of degree at least five, J. Austral. Math. Soc. (Series A) 30 (1980), 37–49. · Zbl 0456.16032 · doi:10.1017/S1446788700021893
[12] A. Fröhlich, The near-ring generated by the inner automorphisms of a finite simple group, J. London Math. Soc. 33 (1958), 95–107. · Zbl 0084.26202
[13] N. Groenewald, A characterization of semiprime ideals in near-rings, J. Austral. Math. Soc. (Series A) 35 (1983), 194–196. · Zbl 0521.16030 · doi:10.1017/S1446788700025660
[14] N. Groenewald, Note on the completely prime radical in near-rings, in: Near-Rings and Near-Fields (ed. by G. Betsch), Proceedings, Tübingen 1985. (Math. Studies, vol. 137, pp. 97-100) Amsterdam, New York, Oxford, Tokyo, North-Holland, 1987. · Zbl 0622.16015
[15] N. Groenewald, Strongly prime near-rings, Proc. Edinburgh Math. Soc. 31 (1988), 337–343. · Zbl 0682.16028 · doi:10.1017/S0013091500006738
[16] N. Groenewald, The completely prime radical in near-rings, Acta Math. Hung. 51 (1988), 301–305. · Zbl 0655.16025 · doi:10.1007/BF01903337
[17] N. Groenewald, Strongly prime near-rings 2, Comm. Alg. 17 (1989), 735–749. · Zbl 0682.16029 · doi:10.1080/00927878908823754
[18] H. Heatherly, Distributive near-rings, Quart. J. Math. Oxford Ser. (2) 24 (1973), 63–70. · Zbl 0261.16017
[19] H. Heatherly and S. Ligh, Pseudo-distributive near-rings, Bull. Austral. Math. Soc. 12 (1975), 449–456. · Zbl 0295.16015 · doi:10.1017/S0004972700024102
[20] P. Jones, Distributive Near-Rings, Thesis Univ. Southw. Louisiana, Lafayette 1976.
[21] K. Kaarli, On Jacobson type radicals of near-rings, Acta Math. Hung. 50 (1987), 71–78. · Zbl 0644.16027 · doi:10.1007/BF01903365
[22] K. Kaarli and T. Kriis, Prime radical of near-rings, Tartu Riikl. Ül. Toimetised 764 (1987), 23–29. · Zbl 0638.16028
[23] R. Laxton, Prime ideals and the ideal radical of a distributively generated near-ring, Math. Z. 83 (1964), 8–17. · Zbl 0123.00903 · doi:10.1007/BF01111100
[24] S. Ligh and Y. Utumi, Some generalizations of strongly regular near-rings, Math. Japan. 21 (1976), 113–116. · Zbl 0358.16023
[25] C. Lyons and J. Malone, Endomorphism near-rings, Proc. Edinburgh Math. Soc. 17 (1970), 71–78. · Zbl 0203.33601 · doi:10.1017/S0013091500009214
[26] J. Malone, More on groups in which each element commutes with its endomorphic image, Proc. Amer. Math. Soc. 65 (1977), 209–214. · Zbl 0371.20022 · doi:10.1090/S0002-9939-1977-0447351-X
[27] G. Mason, Reflexive ideals, Comm. Alg. 9 (1981), 1709–1724. · Zbl 0468.16024 · doi:10.1080/00927878108822678
[28] J.D.P. Meldrum, Near-Rings and their Links with Groups, Boston London Melbourne, Pitman 1985. · Zbl 0658.16029
[29] A. Oswald, Near-rings in which every N-subgroup is principal, Proc. London Math. Soc. (3) 28 (1974), 67–88. · Zbl 0277.16025
[30] G. Pilz, Near-Rings, 2nd ed., Amsterdam, New York, Oxford, North-Holland, 1983. · Zbl 0521.16028
[31] D. Ramakotaiah, Radicals for near-rings, Math. Z. 97 (1967), 45–56. · Zbl 0145.25503 · doi:10.1007/BF01111122
[32] D. Ramakotaiah and G. Koteswara Rao, IFP near-rings, J. Austral. Math. Soc. (Series A) 27 (1979), 365–370. · Zbl 0409.16029 · doi:10.1017/S1446788700012477
[33] V. Sambasiva Rao, A characterization of semiprime ideals in near-rings, J. Austral. Math. Soc. (Series A) 32 (1982), 212–214. · Zbl 0491.16033 · doi:10.1017/S1446788700024551
[34] V. Sambasiva Rao and Bh. Satyanarayana, The prime radical in near-rings, Indian J. Pure Appl. Math. 15 (1984), 361–364. · Zbl 0533.16020
[35] Y.V. Reddy and C.V.L.N. Murty, Semi-symmetric ideals in near-rings, Indian J. Pure Appl. Math. 16 (1985), 17–21. · Zbl 0584.16022
[36] L. Rédei, Das ”schiefe Produkt” in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungszahlen, zu denen nur kommutative Gruppen gehören, Comment. Math. Helv. 20 (1947), 225–264. · Zbl 0035.01503 · doi:10.1007/BF02568131
[37] R. Scapellato, On geometric near-rings in: Near-Rings and Near-Fields (ed. by G. Betsch), Proceedings, Tübingen 1985. (Math. Studies, vol. 137, pp. 253-254) Amsterdam, New York, Oxford, Tokyo, North-Holland, 1987. · Zbl 0618.16030
[38] A. Van der Walt, Prime ideals and nil radicals in near-rings, Arch. Math. 15 (1964), 408–414. · Zbl 0125.01001 · doi:10.1007/BF01589223
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.