Abstract
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
Highlights
The formal definition of semirings was introduced by H
In the year 1991, Bresar (9) introduced the concept of generalized derivation in rings as follows; Ahmed et al / Indian Journal of Science and Technology 2020;13(22):2214–2219 an additive mapping G: R → R associated with derivation d: R → R such that G(xy) = G(x)y + xd(y) holds for all x, y ∈ R
We introduce the concept of generalized m-derivations in (1) as an additive map G : R → R on ring R and there exists a multiplicative derivation d of R such that G(xy) = G(x)y +
Summary
The formal definition of semirings was introduced by H. In the year 1991, Bresar (9) introduced the concept of generalized derivation in rings as follows; Ahmed et al / Indian Journal of Science and Technology 2020;13(22):2214–2219 an additive mapping G: R → R associated with derivation d: R → R such that G(xy) = G(x)y + xd(y) holds for all x, y ∈ R. We introduce the concept of generalized m-derivations in (1) as an additive map G : R → R on ring R and there exists a multiplicative derivation d of R such that G(xy) = G(x)y +. A semiring S is said to be weakly left cancellative (WLC) if axb = axc for all x ∈ S, implies either a = 0 or b = c This notion is recently introduced by V. G is multiplicative derivation which is non-additive and G is generalized m-derivations
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
