Primitive hyperideals and hyperstructure spaces of hyperrings

Document Type : Research Paper

Authors

1 Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7600, South Africa. National Institute for Theoretical and Computational Sciences (NITheCS), South Africa.

2 Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa. National Institute for Theoretical and Computational Sciences (NITheCS), South Africa.

3 Department of Mathematical Sciences, Yazd University, Yazd, Iran.

10.48308/cgasa.2023.234185.1460

Abstract

We introduce primitive hyperideals of a hyperring $R$ and show how they are related to $R$ itself, and to maximal and prime hyperideals of $R$. We endow a Jacobson topology on the set of primitive hyperideals of $R$ and study the topological properties of the corresponding hyperstructure space.

Keywords

Main Subjects


[1] Bordbar, H. and Cristea, I., About the normal projectivity and injectivity of Krasner hypermodules, Axioms 10(2) (2021), 83.
[2] Bordbar, H., Novak, M., and Cristea, I., A note on the support of a hypermodule, J. Algebra Appl. 19 (2020) 2050019.
[3] Connes, A. and Consani, C., The hyperring of ad´ele classes, J. Number Theory 131(2) (2011), 159-194.
[4] Connes, A. and Consani, C., From monoids to hyperstructures: in search of an absolute arithmetic, in: Casimir Force, “Casimir Operators and the Riemann Hypothesis”, de Gruyter, (2010), 147-198.
[5] Corsini, P. and Leoreanu-Fotea, V., “Applications of Hyperstructure Theory”, Vol. 5, Springer, 2003.
[6] Davvaz, B. and Salasi, A., A realization of hyperrings, Comm. Algebra 34(12) (2006), 4389-4400.
[7] Davvaz, B. and Leoreanu-Fotea, V., “Hyperring Theory and Applications”, International Academic Press, 2007.
[8] Dixmier, J., “Enveloping Algebras”, Amer. Math. Soc., 1996.
[9] Hamzekolaee, A.R.M., Norouzi, M., and Leoreanu-Fotea, V., A new approach to smallness in hypermodules, Algebr. Struct. their Appl., 8(1) (2021), 131-145.
[10] Irving, R.S., Prime Ideals of Ore extensions over commutative rings, J. Algebra 56 (1979), 315-342.
[11] Jacobson, N.,A topology for the set of primitive ideals in an arbitrary ring, Proc. Nat. Acad. Sei. U.S.A. 31 (1945), 333-338.
[12] Jacobson, N., “Structure of Rings”, Amer. Math. Soc. Colloquium Publications, Vol. 37, Providence, 1956.
[13] Jacobson, N., “PI-algebras. An Introduction”, Springer-Verlag, 1975.
[14] Joseph, A., Primitive ideals in enveloping algebras, Proc. ICM Warsaw (1983), 403-414.
[15] Joseph, A., “Quantum Groups and their Primitive Ideals”, Springer, 1995.
[16] Jun, J., Algebraic geometry over hyperrings, Adv. Math., 323 (2018), 142-192.
[17] Krasner, M., A class of hyperrings and hyperfields, Internat. J. Math. Math. Sci. 6(2) (1983), 307-311.
[18] Kucherov, A.A., Pikhtilkova, O.A., and Pikhtilkov, S.A., On primitive Lie algebras, J. Math. Sci. 186(4) (2012), 651-654.
[19] Marty, F., Sur une g´en´eralization de la notion de groupe, in: 8th Congress Math. Scandinaves (1934), 45-49.
[20] Massouros, Ch.G., Free and cyclic hypermodules, Ann. Mat. Pura Appl. 150(4) (1988), 153-66.
[21] Mahjoob, R. and Ghaffari, V., Zariski topology for second subhypermodules, Ital. J. Pure Appl. Math. 39 (2018), 554-568.
[22] Omidi, S. and Davvaz, B., Hyperideal theory in ordered Krasner hyperrings, Analele Univ. “Ovidius” din Constanta, Math. Series 27(1) (2019), 193-210.
[23] Rowen, L.H., “Ring Theory”, Vol. I, Academic Press, 1988.
[24] Siraworakun, A, “Some Properties of Hypermodules over Krasner Hyperrings”, Chulalongkorn University, 2007.
[25] T¨urkmen, B. N., Bordbar, H,, and Cristea, I., Supplements related to normal π-projective hypermodules, Mathematics 10, 1945 (2022), (15 pages).