Additivity of multiplicative (generalized) skew semi-derivations on rings
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Sk Aziz
Abstract
In this paper, we introduce a new class of derivations that generalizes skew derivations and semi-derivations, and we call it skew semi-derivation. Furthermore, we present a study of the conditions under which this type of multiplicative derivation becomes additive.
Funding source: University Grants Commission
Award Identifier / Grant number: 1256/(CSIR-UGC NET JUNE 2019)
Funding statement: The first author is thankful to the University Grants Commission (UGC), Government of India for financial supports under File No. 16-9(June 2019)/2019 (NET/CSIR), UGC Ref. No. 1256/(CSIR-UGC NET JUNE 2019) dated 16/12/2019.
Acknowledgements
The authors would like to thank the anonymous referee(s) for their valuable comments to improve the presentation of the paper.
References
[1] S. Aziz, A. Ghosh and O. Prakash, Additivity of multiplicative (generalized) maps over rings, preprint (2023), https://arxiv.org/abs/2302.14571. Search in Google Scholar
[2] S. Aziz, A. Ghosh and O. Prakash, Some generalized Jordan maps on triangular rings force additivity, preprint (2023), https://arxiv.org/abs/2301.08093. 10.24330/ieja.1488471Search in Google Scholar
[3] J. Bergen, Derivations in prime rings, Canad. Math. Bull. 26 (1983), no. 3, 267–270. 10.4153/CMB-1983-042-2Search in Google Scholar
[4] M. Brešar, Semiderivations of prime rings, Proc. Amer. Math. Soc. 108 (1990), no. 4, 859–860. 10.1090/S0002-9939-1990-1007488-XSearch in Google Scholar
[5] J. C. Chang, On semiderivations of prime rings, Chinese J. Math. 12 (1984), no. 4, 255–262. Search in Google Scholar
[6] M.-C. Chou and C.-K. Liu, An Engel condition with skew derivations, Monatsh. Math. 158 (2009), no. 3, 259–270. 10.1007/s00605-008-0043-5Search in Google Scholar
[7] C.-L. Chuang, Identities with skew derivations, J. Algebra 224 (2000), no. 2, 292–335. 10.1006/jabr.1999.8052Search in Google Scholar
[8] M. N. Daif, When is a multiplicative derivation additive?, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 615–618. 10.1155/S0161171291000844Search in Google Scholar
[9] T. El-Sayiad, M. N. Daif and V. De Filippis, Multiplicativity of left centralizers forcing additivity, Bol. Soc. Parana. Mat. (3) 32 (2014), no. 1, 61–69. 10.5269/bspm.v32i1.17274Search in Google Scholar
[10] B. L. M. Ferreira, Multiplicative maps on triangular n-matrix rings, Int. J. Math. Game Theory Algebr. 23 (2014), no. 2, 1–14. Search in Google Scholar
[11] N. Jacobson, Abstract derivation and Lie algebras, Trans. Amer. Math. Soc. 42 (1937), no. 2, 206–224. 10.1090/S0002-9947-1937-1501922-7Search in Google Scholar
[12] R. E. Johnson, Rings with unique addition, Proc. Amer. Math. Soc. 9 (1958), 57–61. 10.1090/S0002-9939-1958-0095868-0Search in Google Scholar
[13] R. V. Kadison and J. R. Ringrose, Derivations and automorphisms of operator algebras, Comm. Math. Phys. 4 (1967), 32–63. 10.1007/BF01645176Search in Google Scholar
[14] V. K. Kharchenko and A. Z. Popov, Skew derivations of prime rings, Comm. Algebra 20 (1992), no. 11, 3321–3345. 10.1080/00927879208824517Search in Google Scholar
[15] A. Leroy, Dérivées logarithmiques pour une 𝑆-dérivation algébrique, Comm. Algebra 13 (1985), no. 1, 85–99. 10.1080/00927878508823149Search in Google Scholar
[16] W. S. Martindale, III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc. 21 (1969), 695–698. 10.1090/S0002-9939-1969-0240129-7Search in Google Scholar
[17] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100. 10.1090/S0002-9939-1957-0095863-0Search in Google Scholar
[18] C. E. Rickart, One-to-one mappings of rings and lattices, Bull. Amer. Math. Soc. 54 (1948), 758–764. 10.1090/S0002-9904-1948-09073-5Search in Google Scholar
[19] M. A. Siddeeque and N. Khan, When are multiplicative semi-derivations additive?, Georgian Math. J. 29 (2022), no. 1, 123–137. 10.1515/gmj-2021-2121Search in Google Scholar
[20] J. Vukman, Derivations on semiprime rings, Bull. Austral. Math. Soc. 53 (1996), no. 3, 353–359. 10.1017/S000497270001710XSearch in Google Scholar
[21] Y. Wang, The additivity of multiplicative maps on rings, Comm. Algebra 37 (2009), no. 7, 2351–2356. 10.1080/00927870802623369Search in Google Scholar
[22] Y. Wang, Additivity of multiplicative maps on triangular rings, Linear Algebra Appl. 434 (2011), no. 3, 625–635. 10.1016/j.laa.2010.09.015Search in Google Scholar
[23] M. Weisfeld, On derivations in division rings, Pacific J. Math. 10 (1960), 335–343. 10.2140/pjm.1960.10.335Search in Google Scholar
[24] V. K. Yadav and R. K. Sharma, Additivity of multiplicative generalized Jordan derivations on rings, Palest. J. Math. 6 (2017), 253–258. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Additivity of multiplicative (generalized) skew semi-derivations on rings
- On statistical convergence of order α in partial metric spaces
- Generalized derivations over amalgamated algebras along an ideal
- Numerical radii of operator matrices in terms of certain complex combinations of operators
- Centralizing identities involving generalized derivations in prime rings
- Two presentations of a weak type inequality for geometric maximal operators
- Existence and exponential stability of solutions for a Balakrishnan–Taylor quasilinear wave equation with strong damping and localized nonlinear damping
- Identities with generalized derivations on Lie ideals and Banach algebras
- Essential norm of Riemann–Stieltjes operator on weighted Bergman spaces with doubling weights
- Concerning the Nakayama property of a module
- Multiplicity result for a (p(x),q(x))-Laplacian-like system with indefinite weights
- Finite groups in which some particular invariant subgroups are TI-subgroups or subnormal subgroups
- The Dirichlet problem in an infinite layer for a system of differential equations with shifts
- Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring ℤ𝑛
- A new fuzzy approach of vehicle routing problem for disaster-stricken zones
Articles in the same Issue
- Frontmatter
- Additivity of multiplicative (generalized) skew semi-derivations on rings
- On statistical convergence of order α in partial metric spaces
- Generalized derivations over amalgamated algebras along an ideal
- Numerical radii of operator matrices in terms of certain complex combinations of operators
- Centralizing identities involving generalized derivations in prime rings
- Two presentations of a weak type inequality for geometric maximal operators
- Existence and exponential stability of solutions for a Balakrishnan–Taylor quasilinear wave equation with strong damping and localized nonlinear damping
- Identities with generalized derivations on Lie ideals and Banach algebras
- Essential norm of Riemann–Stieltjes operator on weighted Bergman spaces with doubling weights
- Concerning the Nakayama property of a module
- Multiplicity result for a (p(x),q(x))-Laplacian-like system with indefinite weights
- Finite groups in which some particular invariant subgroups are TI-subgroups or subnormal subgroups
- The Dirichlet problem in an infinite layer for a system of differential equations with shifts
- Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring ℤ𝑛
- A new fuzzy approach of vehicle routing problem for disaster-stricken zones
