On prime strong ideals of a seminearring. (English) Zbl 1474.16125

Summary: The concepts of prime ideals and corresponding radicals play an important role in the study of nearrings. In this paper, we define different prime strong ideals of a seminearring \(S\) and study the corresponding prime radicals. In particular, we prove that \(P_e=\{S\mid P_e(S)=S\}\) is a Kurosh-Amitsur radical class where \(P_e(S)\) denotes the intersection of equiprime strong ideals of \(S\).


16Y30 Near-rings
16Y60 Semirings
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[1] J. Ahsan,Seminear-rings characterized by their S-ideals. I, Proc. Japan Acad. Ser. A,71(5) (1995), 101-103. · Zbl 0842.16035
[2] J. Ahsan,Seminear-rings characterized by their S-ideals. II, Proc. Japan Acad. Ser. A,71(6) (1995), 111-113. · Zbl 0842.16036
[3] T. Anderson, K. Kaarli, R. Wiegandt,Radicals and subdirect decompositions, Comm. Algebra, 13(2)(1985), 479-494. · Zbl 0553.16004
[4] M. Bataineh, R. Malas,Generalizations of prime ideals over commutative semirings, Math. Vesnik.,66(2)(2014), 133-139. · Zbl 1458.16051
[5] G. Birkenmeier, H. Heatherly, E. Lee,Completely prime ideals and radicals in nearrings in Near-Rings and Near-Fields, Springer publishers, 1995. · Zbl 0839.16040
[6] G.L. Booth, N.J. Groenewald, S. Veldsman,A Kurosh-Amitsur prime radical for near-rings, Comm. Algebra,18(9)(1990), 3111-3122. · Zbl 0706.16025
[7] N.J. Divinsky,Rings and radicals, University of Toronto Press, 1965. · Zbl 0138.26303
[8] M. K. Dubey, P. Sarohe,On(n−1, n)-φ-prime ideals in semirings, Mat. Vesnik.,67(3)(2015), 222-232. · Zbl 1474.16130
[9] B.J. Gardner, R. Wiegandt,Radical theory of rings, CRC Press, 2003. · Zbl 1034.16025
[10] J.S. Golan,Semirings and their Applications, Kluwer Acadamic Publishers, 1999. · Zbl 0947.16034
[11] N.J. Groenewald,The completely prime radical in near-rings, Acta Math. Hung.,51(3-4) (1988), 301-305. · Zbl 0655.16025
[12] B. Jagadeesha, S.P. Kuncham, B.S. Kedukodi,Implications on a Lattice, Fuzzy. Inf. Eng., 8(4)(2016), 411-425.
[13] K. Koppula, B.S. Kedukodi, S.P. Kuncham,On strong ideals of seminearrings, (Communicated).
[14] K.V. Krishna, N. Chatterjee,A necessary condition to test the minimality of generalized linear sequential machines using the theory of near-semirings, Algebra Discrete Math.,4(3)(2005), 30-45. · Zbl 1093.16038
[15] K.V. Krishna, N. Chatterjee,Representation of near-semirings and approximation of their categories, Southeast Asian Bull. Math.,31(2007), 903-914. · Zbl 1140.16309
[16] S.P. Kuncham, B. Jagadeesha, B.S. Kedukodi,Interval valued L-fuzzy cosets of nearrings and isomorphism theorems, Afr. Mat.,27(3)(2016), 393-408. · Zbl 1377.16041
[17] H. Nayak, S.P. Kuncham, B.S. Kedukodi,Extensions of boolean rings and nearrings, Journal of Siberian Federal University - Mathematics and Physics,12(1)(2019), 58-67. · Zbl 07325480
[18] G. Pilz,Near-rings: The Theory and Its Applications, Revised edition, North Hollond, 1983. · Zbl 0521.16028
[19] R.S. Rao, K.S. Prasad,A Kurosh-Amitsur left jacobson radical for right near-rings, Bull. Korean Math. Soc.,45(2008), 457-466. · Zbl 1158.16022
[20] W.G. Van Hoorn, B. Van Rootselaar,Fundamental notions in the theory of seminearrings, Compos. Math.,18(1967), 65-78. · Zbl 0166.03904
[21] S. Veldsman,Modulo-constant ideal-hereditary radicals of nearrings, Quaest. Math.,11 (1988), 253-278. · Zbl 0656.16017
[22] S. Veldsman,On equiprime near-rings, Commun. Algebra.,20(1992), 2569-2587. · Zbl 0795.16034
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