Abstract
The main goal of this paper is to characterize Jordan two-sided centralizers, Jordan centralizers and related maps on triangular rings without identity. As an application of our main theorem, we characterize Jordan generalized derivations on triangular rings. Precisely, we prove that every Jordan generalized derivation on a triangular ring is a two-sided generalized derivation. As consequences, and apart from proving the other results, many known theorems can be either generalized or deduced.
Funding statement: The research of the second named author is supported by SERB-DST MATRICS Project under Grant No. MTR/2019/000603, India.
References
[1] A. Aboubakr and S. González, Orthogonality of two left and right generalized derivations on ideals in semiprime rings, Rend. Circ. Mat. Palermo (2) 68 (2019), no. 3, 611–620. 10.1007/s12215-018-0383-5Search in Google Scholar
[2] S. Ali and C. Haetinger, Jordan α-centralizers in rings and some applications, Bol. Soc. Parana. Mat. (3) 26 (2008), no. 1–2, 71–80. 10.5269/bspm.v26i1-2.7405Search in Google Scholar
[3] M. Brešar, On the distance of the composition of two derivations to the generalized derivations, Glasg. Math. J. 33 (1991), no. 1, 89–93. 10.1017/S0017089500008077Search in Google Scholar
[4] Q. Chen, X. Fang and C. Li, The characterization of generalized Jordan centralizers on algebras, Bol. Soc. Parana. Mat. (3) 35 (2017), no. 3, 225–240. 10.5269/bspm.v35i3.30008Search in Google Scholar
[5] Q. Chen, X. Fang and C. Li, The characterization of generalized Jordan centralizers on triangular algebras, J. Funct. Spaces 2018 (2018), Article ID 6037615. 10.1155/2018/6037615Search in Google Scholar
[6] A. Fošner and W. Jing, A note on Jordan derivations of triangular rings, Aequationes Math. 94 (2020), no. 2, 277–285. 10.1007/s00010-019-00689-ySearch in Google Scholar
[7] M. Fošner and N. Peršin, On certain functional equation related to two-sided centralizers, Aequationes Math. 85 (2013), no. 3, 329–346. 10.1007/s00010-012-0144-zSearch in Google Scholar
[8] H. Ghahramani, Characterizing Jordan maps on triangular rings through commutative zero products, Mediterr. J. Math. 15 (2018), no. 2, Paper No. 38. 10.1007/s00009-018-1082-3Search in Google Scholar
[9] J. He, J. Li and W. Qian, Characterizations of centralizers and derivations on some algebras, J. Korean Math. Soc. 54 (2017), no. 2, 685–696. 10.4134/JKMS.j160265Search in Google Scholar
[10] A. Hosseini, Characterization of two-sided generalized derivations, Acta Sci. Math. (Szeged) 86 (2020), no. 3–4, 577–600. 10.14232/actasm-020-295-8Search in Google Scholar
[11] W. Jing, Additivity of Lie centralizers on triangular rings, Math and Computer Science Working Papers 8, Fayetteville State University, 2011, https://digitalcommons.uncfsu.edu/cgi/viewcontent.cgi?article=1008&context=macsc_wp. Search in Google Scholar
[12] W. Jing and F. Lu, Additivity of Jordan (triple) derivations on rings, Comm. Algebra 40 (2012), no. 8, 2700–2719. 10.1080/00927872.2011.584927Search in Google Scholar
[13] I. Kosi-Ulbl, On a functional equation related to two-sided centralizers, Ann. Math. Sil. 32 (2018), no. 1, 227–235. 10.1515/amsil-2017-0014Search in Google Scholar
[14]
I. Kosi-Ulbl and J. Vukman,
On centralizers of standard operator algebras and semisimple
[15] T.-K. Lee and T. C. Quynh, Centralizers and Jordan triple derivations of semiprime rings, Comm. Algebra 47 (2019), no. 1, 236–251. 10.1080/00927872.2018.1472275Search in Google Scholar
[16]
J. Li, Q. Shen and J. Guo,
On generalized
[17] L. Liu, On Jordan centralizers of triangular algebras, Banach J. Math. Anal. 10 (2016), no. 2, 223–234. 10.1215/17358787-3492545Search in Google Scholar
[18] L. Liu, On nonlinear Lie centralizers of generalized matrix algebras, Linear Multilinear Algebra 70 (2022), no. 14, 2693–2705. 10.1080/03081087.2020.1810605Search in Google Scholar
[19] X. F. Qi and J. C. Hou, Characterizing centralizers and generalized derivations on triangular algebras by acting on zero product, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 7, 1245–1256. 10.1007/s10114-013-2068-5Search in Google Scholar
[20] J. Vukman, An identity related to centralizers in semiprime rings, Comment. Math. Univ. Carolin. 40 (1999), no. 3, 447–456. Search in Google Scholar
[21] J. Vukman, Identities related to derivations and centralizers on standard operator algebras, Taiwanese J. Math. 11 (2007), no. 1, 255–265. 10.11650/twjm/1500404650Search in Google Scholar
[22]
J. Vukman,
On
[23] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32 (1991), no. 4, 609–614. Search in Google Scholar
[24] J.-H. Zhang and W.-Y. Yu, Jordan derivations of triangular algebras, Linear Algebra Appl. 419 (2006), no. 1, 251–255. 10.1016/j.laa.2006.04.015Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- On the numerical solvability of the initial problem with weight for ordinary linear differential systems with singularities
- Multiplicative Lie-type derivations on standard operator algebras
- Numerical radius inequalities and estimation of zeros of polynomials
- Convolution equations on the Lie group G = (-1,1)
- Positive solutions for a fourth-order p-Laplacian boundary value problem
- Existence and asymptotic behavior of strictly convex solutions for singular k-Hessian equations with nonlinear gradient terms
- Characterization of Jordan two-sided centralizers and related maps on triangular rings
- Convergence and integrability of rational and double rational trigonometric series with coefficients of bounded variation of higher order
- Weak and strong type inequalities criteria for fractional maximal functions and fractional integrals associated with Gegenbauer differential operator
- Generating sets of F/R' Leibniz algebras
- New results of uncertain integrals and applications
- Multilinear commutators of multilinear strongly singular integral operators with generalized kernels
Articles in the same Issue
- Frontmatter
- On the numerical solvability of the initial problem with weight for ordinary linear differential systems with singularities
- Multiplicative Lie-type derivations on standard operator algebras
- Numerical radius inequalities and estimation of zeros of polynomials
- Convolution equations on the Lie group G = (-1,1)
- Positive solutions for a fourth-order p-Laplacian boundary value problem
- Existence and asymptotic behavior of strictly convex solutions for singular k-Hessian equations with nonlinear gradient terms
- Characterization of Jordan two-sided centralizers and related maps on triangular rings
- Convergence and integrability of rational and double rational trigonometric series with coefficients of bounded variation of higher order
- Weak and strong type inequalities criteria for fractional maximal functions and fractional integrals associated with Gegenbauer differential operator
- Generating sets of F/R' Leibniz algebras
- New results of uncertain integrals and applications
- Multilinear commutators of multilinear strongly singular integral operators with generalized kernels