Abstract
In this paper, we introduce the concept of clean semiring as a generalization of clean ring. A semiring is said to be clean if its every nonzero element can be written as the sum of an idempotent and a unit. We study the notion of clean semiring and obtain some important characterizations of clean semiring. We also study the notion of exchange semiring and find out the connenction between clean semiring and exchange semiring.
Similar content being viewed by others
References
Ahn, M.S., Anderson, D.D.: Weakly clean rings and almost clean rings. Rocky Mt. J. Math. 36(3), 783–798 (2006)
Ashrafi, N., Nasibi, E. : \(r\)-Clean rings. Math. Rep. 15(65)(2), 125–132 (2013)
Chen, J., Wang, Z., Zhou, Y.: Rings in which elements are uniquely the sum of an idempotent and a unit that commute. J. Pure Appl. Algebra 213, 215–223 (2009)
Golan, J.S.: Semirings and Their Applications. Springer, Dordrecht (1999)
Hebisch, U., Weinert, H.J.: Algebric Theory and Applications Computer Science. World Scientific, Singapore (1998)
Kosan, T., Sahinkaya, S., Zhou, Y.: On weakly clean rings. Commun. Algebra 45(8), 3494–3502 (2017)
McGovern, W.W.: Neat rings. J. Pure Appl. Algebra 205(2), 243–265 (2006)
Nicholson, W.K.: Lifting idempotents and exchange rings. Trans. Am. Math. Soc. 229, 269–278 (1977)
Nicholson, W.K.: Strongly clean rings and Fitting’s lemma. Commun. Algebra 27(8), 3583–3592 (1999)
Nicholson, W.K., Zhou, Y.: Clean general rings. J. Algebra 291, 297–311 (2005)
Purkait, S.: On strongly m-clean rings and m-semiperfect rings. Commun. Algebra 48(10), 4531–4541 (2020)
Purkait, S., Dutta, T.K., Kar, S.: On m-clean and strongly m-clean rings. Commun. Algebra 48(1), 218–227 (2020)
Warfield, R.B.: Exchange rings and decomposition of modules. Math. Ann. 199, 31–36 (1972)
Yang, X., Zhou, Y.: Some families of strongly clean rings. Linear Algebra Appl. 425, 119–129 (2007)
Acknowledgements
D. Das is grateful to UGC, India for providing financial support as Junior Research Fellow (JRF). The authors are thankful to the anonymous referee for making some useful comments and suggestions which have definitely improved the paper.
Author information
Authors and Affiliations
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kar, S., Das, D. Clean semiring. Beitr Algebra Geom 64, 197–207 (2023). https://doi.org/10.1007/s13366-022-00628-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-022-00628-0