Abstract
We study left pure ideals, left pure k-ideals, right pure ideals and right pure k-ideals of semirings, introduce the notions of quasi-pure ideals, quasi-pure k-ideals, bi-pure ideals and bi-pure k-ideals of semirings and characterize regular and k-regular semirings using their pure ideals and their pure k-ideals, respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adhikari, M.R., Sen, M.K., Weinert, H.J.: On \(k\)-regular semirings. Bull. Calcutta Math. Soc. 88, 141–144 (1996)
Ahsan, J., Takahashi, M.: Pure spectrum of a monoid with zero. Kobe J. Math. 6, 163–182 (1989)
Bhuniya, A.K., Jana, K.: Bi-ideals in \(k\)-regular and intra \(k\)-regular semirings. Discussiones Mathematicae General Algebra and Applications. 31(1), 5–23 (2011)
Bourne, S.: The Jacobson radical of a semiring. Proc. Nati. Acad. Sci. USA 31, 163–170 (1951)
Hebisch, U., Weinert, H. J.: Semirings Algebraic Theory and Applications in Computer Science Vol.5 of Series in Algebra. Singapore: World Scientific Publishing (1998)
Henriksen, M.: Ideals in semirings with commutative addition. Notices Am. Math. Soc. 6, 321 (1958)
Jagatap, R.D.: Right \(k\)-weakly regular \(\Gamma \)-semirings. Algebra 2014, 948032 (2014)
Jana, K.: Quasi \(k\)-ideals in \(k\)-regular and intra \(k\)-regular semirings. Pure. Math. Appl. 22, 65–74 (2011)
Lajos, S., Szasz, F.A.: On the bi-ideals in associative ring. Proc. Jpn. Acad. 46(6), 505–507 (1970)
Palakawong na Ayutthaya, P., Pibaljommee, B.: Characterizations of ordered \(k\)-regular semirings by ordered quasi \(k\)-ideals. Quasigroups Related Systems 25(1), 109-120 (2017)
Palakawong na Ayutthaya, P., Pibaljommee, B.: Characterizations of ordered intra \(k\)-regular semirings by ordered \(k\)-ideals. Commun. Korean Math. Soc. 33(1) , 1-12 (2018)
Patchakhieo, S., Pibaljommee, B.: Characterizations of ordered \(k\)-regular semirings by ordered \(k\)-ideals. Asian-Euro. J. Math. 10, 1750020 (2017)
Senarat, P., Pibaljommee, B.: Prime ordered \(k\)-bi-ideals in ordered semirings. Quasigroups Related Syst. 25(1), 121–132 (2017)
Shabir, M., Iqbal, M.S.: One sided prime ideals in semirings. Kyungpook Math. J. 47(4), 473–480 (2007)
Steinfeld, O.: Quasi-ideals in rings and semigroups. Budapest, AkadémiaiKiadó (1978)
Vandiver, H.S.: Note on a simple type of algebra in which the cancellation law of addition does not hold. Bull. Am. Math. Soc. 40, 914–920 (1978)
von Neumann, J.: On regular rings. Proc. Natl. Acad. Sci. USA 22, 707–713 (1936)
Weinert, H.J., Sen, M.K., Adhikari, M.R.: One-sided \(k\)-ideals and \(h\)-ideals in semirings. Mathematica Pannonica 7(1), 147–162 (1996)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Palakawong na Ayutthaya, P., Pibaljommee, B. & Pinakano, P. Characterizations of regular and k-regular semirings by pure ideals and pure k-ideals. Afr. Mat. 34, 20 (2023). https://doi.org/10.1007/s13370-023-01057-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13370-023-01057-5