Geodesic
Reverse derivations and generalized reverse derivations in semirings
Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 217-232.

Voir la notice de l'article provenant de la source Library of Science

In this article we extend the results on reverse derivation in rings to semirings. First we dispose of reverse derivations in prime semirings analogous to Herstein's result [7]. Then, we prove that reverse derivation is just an ordinary derivation in semiprime semirings if and only if it is a central derivation. We also define generalized reverse derivations and obtain some commutativity results which extend the results in [11]. The primary technique we use in these results is the use of derivations and reverse derivations in ring of differences R^Δ corresponding to the semiring R and the fact that R is embedded in R^Δ. This fact allows us to travel back and forth between R and R^Δ and serve as a key tool in obtaining the desired results.
Keywords: Reverse derivations, derivations, semirings, l-generalized reverse derivations, r-generalized reverse derivations, generalized reverse derivations and semiprime semirings 
@article{DMGAA_2024_44_1_a13,
     author = {Swaminathan, Ganesh and Venkatachalam, Selvan},
     title = {Reverse derivations and generalized reverse derivations in semirings},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {217--232},
     publisher = {mathdoc},
     volume = {44},
     number = {1},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a13/}
}
TY  - JOUR
AU  - Swaminathan, Ganesh
AU  - Venkatachalam, Selvan
TI  - Reverse derivations and generalized reverse derivations in semirings
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2024
SP  - 217
EP  - 232
VL  - 44
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a13/
LA  - en
ID  - DMGAA_2024_44_1_a13
ER  - 
%0 Journal Article
%A Swaminathan, Ganesh
%A Venkatachalam, Selvan
%T Reverse derivations and generalized reverse derivations in semirings
%J Discussiones Mathematicae. General Algebra and Applications
%D 2024
%P 217-232
%V 44
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a13/
%G en
%F DMGAA_2024_44_1_a13
Swaminathan, Ganesh; Venkatachalam, Selvan. Reverse derivations and generalized reverse derivations in semirings. Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 217-232. http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a13/

[1] A. Aboubakr and S. González, Generalized reverse derivation on semiprime rings, Siberian Math. J. 56 (2015) 199–205. https://doi.org/10.1134/S0037446615020019

[2] Y. Ahmed and W. Dudek, On Generalised Reverse Derivations in Semirings, Bull. Iran. Math. Soc. 48 (2022) 895–904. https://doi.org/10.1007/s41980-021-00552-4

[3] S.I. Dimitrov, Derivations on semirings, Appl. Math. in Eng. and Econ. – 43th. Int. Conf., AIP Conf. Proc. 1910 (2017) 060011. https://doi.org/10.1063/1.5014005

[4] S. Ganesh and V. Selvan, Posner’s theorems in semirings, Indian J. Pure Appl. Math., in press. https://doi.org/10.1007/s13226-023-00388-0

[5] S. Ganesh and V. Selvan, Jordan structures in semirings, Rend. Circ. Mat. Palermo, Ser. II 72 (2022) 3587–3596. https://doi.org/10.1007/s12215-022-00838-4

[6] J.S. Golan, Semirings and Their Applications (Kluwer Academic Publishers, 1999).

[7] I.N. Herstein, Jordan Derivations of Prime Rings, Proc. Amer. Math. Soc. 8 (1957) 1104–1110. https://doi.org/10.1090/S0002-9939-1957-0095864-2

[8] I.N. Herstein, Rings with involution (The University of Chicago Press, 1976).

[9] M. Samman and N. Alyamani, Derivations and Reverse Derivations in Semiprime Rings, Internat. Math. Forum 2 (2007) 1895–1902. https://doi.org/10.12988/imf.2007.07168

[10] N. Sugantha Meena and M. Chandramouleeswaran, Reverse derivation on semirings, Int. J. Pure and Appl. Math. 104 (2015) 203–212. https://doi.org/10.12732/ijpam.v104i2.5

[11] S.K. Tiwari, R.K. Sharma and B. Dhara, Some theorems of commutativity on semiprime rings with mappings, South. Asian Bull. Math. 42 (2018) 279–292.