$k$-Congruences and the Zariski topology in semirings

Abstract

The purpose of this paper is to study topological properties of both the set of all $k$-prime ideals and the set of all $k$-prime congruences for any commutative semiring with zero and identity. We first prove that the $k$-prime spectrum, i.e. the set of all $k$-prime ideals equipped with the Zariski topology is a spectral space, and then prove that the set of all $k$-prime congruences is homeomorphic to the $k$-prime spectrum with respect to their Zariski topologies.

Keywords

• semiring
• ideal
• congruence
• prime ideal
• Zariski topology

References

1. [1] S.E. Atani and R.E. Atani, A Zariski topology for k-semirings, Quasigroups Related Systems 20, 29-36, 2012.
2. [2] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison Wesley, Massachusetts, 1969.
3. [3] F. Callialp, G. Ulucak and U. Tekir, On the Zariski topology over an L-module M, Turkish J. Math. 41, 326-336, 2017.
4. [4] R. El Bashir and T. Kepka, Congruence-simple semirings, Semigroup Forum 75, 588-608, 2007.
5. [5] J.S. Golan, Semirings and their Applications, Kluwer Academic Publishers, Dordrecht, 1999.
6. [6] S.C. Han, Maximal k-ideals and r-ideals in semirings, J. Algebra Appl. 14 (10), 1250195, 13 pages, 2015.
7. [7] U. Hebisch and H.J. Weinert, Semirings: Algebraic Theory and Applications in Computer Science, World Scientific, Singapore, 1998.
8. [8] M. Henriksen, Ideals in semirings with commutative addition, Amer. Math. Soc. Notices 5, 321, 1958.

9. [9] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142, 43-60, 1969.
10. [10] P. Lescot, Absolute algebra II – Ideals and spectra, J. Pure Appl. Algebra 215, 1782- 1790, 2011.
11. [11] P. Lescot, Absolute algebra III – The saturated spectrum, J. Pure Appl. Algebra 216, 1004-1015, 2012.
12. [12] P. Lescot, Prime and primary ideals in semirings, Osaka J. Math. 52, 721-736, 2015.
13. [13] G. Maze, C. Monico and J. Rosenthal, Public key cryptography based on semigroup actions, Adv. Math. Commun. 1, 489-507, 2007.
14. [14] A. Pena, L.M. Ruza and J. Vielma, Separation axioms and the prime spectrum of commutative semirings, Rev. Notas Mat. 5 (2), 66-82, 2009.
15. [15] M.K. Sen and M.R. Adhikari, On maximal k-ideals of semirings, Proc. Amer. Math. Soc. 118, 699-703, 1993.
16. [16] H.E. Stone, Matrix representation of simple halfrings, Trans. Amer. Math. Soc. 233, 339-353, 1977.
17. [17] H.S. Vandiver, Note on a simple type of algebra in which the cancellation law of addition does not hold, Bull. Amer. Math. Soc. 40, 916-920, 1934.
18. [18] H.J. Weinert, M.K. Sen and M.R. Adhikari, One-sided k-ideals and h-ideals in semirings, Math. Pannon. 7 (1), 147-162, 1996.
19. [19] G. Yesilot, On prime and maximal k-subsemimodules of semimodules, Hacet. J. Math. Stat. 39, 305-312, 2010.
20. [20] B. Zhou and W. Yao, Relations between ideals and regular congruences in idempotent semirings with a zero, Basic Sci. J. Textile Univ. 24, 253-255, 2011.