• P-ISSN 0974-6846 E-ISSN 0974-5645

Indian Journal of Science and Technology


Indian Journal of Science and Technology

Year: 2020, Volume: 13, Issue: 22, Pages: 2214-2219

Original Article

A note on generalized m-derivations to weakly cancellative semirings

Received Date:23 May 2020, Accepted Date:01 June 2020, Published Date:25 June 2020


Objectives: Semirings is an important algebraic structure with applications in theory of automata, formal languages and theoretical computer science. The mappings which enforces commutativity in semirings remains attractive for researchers, since commutativity would be helpful in calculations and bring it's applications to ease. Our aim is to enforce commutativity in semirings by generalizing the classical theorem of Martindale [14, Theorem 3] with generalized m-derivation. Further, we discuss that composition of two generalized m-derivations ensure that one of their associated derivation must be trivial. Method: We use generalized m-derivations which is associated to multiplicative derivations in certain semirings. Findings: We find the conditions of commutativity in semirings through these particular generalized m-derivations. Moreover, we discuss the characteristics of these mappings in weakly cancellative semirings. Novelty: The concept of generalized m-derivations is newly introduced by us in ring theory in (1) and here we extend this concept to theory of semirings. We attempt to induce commutativity in weakly cancellative semirings (2) whose concept is unorthodox in the theory of semirings. This article pave new ways to study derivations and its applications on semirings.

Keywords: Derivations; generalized m-derivations; weakly cancellative semiring; commutativity 


  1. Ahmed Y, Nadeem M, Aslam M. Posner’s Theorems For Generalized M-Derivations. International Journal of Grid and Distributed Computing. 2020;13(1):898–906.
  2. Filippis VD, Mamouni A, Oukhtite L. Weakly left cancellative semirings with derivations. São Paulo Journal of Mathematical Sciences. 2020;14(1):351–360. Available from: https://dx.doi.org/10.1007/s40863-019-00148-1
  3. Golan JS. Semirings and Their Applications. Dordrecht. Kluwer Academic Publishers. 1999.
  4. Kostolányi P, Mišún F. Alternating weighted automata over commutative semirings. Theoretical Computer Science. 2018;740:1–27. Available from: https://dx.doi.org/10.1016/j.tcs.2018.05.003
  5. Ahmed Y, Dudek WA. Stronger Lie derivations on MA-semirings. Afrika Matematika. 2020;p. 2020. Available from: https://dx.doi.org/10.1007/s13370-020-00768-3
  6. Brešar M. Centralizing mappings and derivations in prime rings. J. Algebra. 1993;156:385–394.
  7. Golbasi O, Koc E. Notes on commutativity of prime rings with generalized derivation. Commun. Fac. Sci. Univ. Ank. Ser. A1-Math. Stat. 2009;58(2):39–46.
  8. Dhara B, Filippis VD. NOTES ON GENERALIZED DERIVATIONS ON LIE IDEALS IN PRIME RINGS. Bulletin of the Korean Mathematical Society. 2009;46(3):599–605. Available from: https://dx.doi.org/10.4134/bkms.2009.46.3.599
  9. Hvala B. Generalized derivations in rings*. Communications in Algebra. 1998;26(4):1147–1166. Available from: https://dx.doi.org/10.1080/00927879808826190
  10. OUKHTITE L, MAMOUNI A. Generalized derivations centralizing on Jordanideals of rings with involution. TURKISH JOURNAL OF MATHEMATICS. 2014;38(2):225–232. Available from: https://dx.doi.org/10.3906/mat-1203-14
  11. Ali L, Khan AM, YA. On Jordan Ideals of Inverse Semirings with Involution. Indian journal of Science andTechnology. 2020;13(04):430–438.
  12. Bell HE, Martindale WS. Centralizing Mappings of Semiprime Rings. Canadian Mathematical Bulletin. 1987;30(1):92–101. Available from: https://dx.doi.org/10.4153/cmb-1987-014-x


© 2020 Ahmed, Aslam, Mahmood. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Published By Indian Society for Education and Environment (iSee)


Subscribe now for latest articles and news.