On skew derivations and antiautomorphisms in prime rings. (English) Zbl 1540.16035
Summary: According to Posner’s second theorem, a prime ring is forced to be commutative if a nonzero centralizing derivation exists on it. In this article, we extend this result to prime rings with antiautomorphisms and nonzero skew derivations. Additionally, a case is shown to demonstrate that the restrictions placed on the theorems’ hypothesis were not unnecessary.
MSC:
| 16W25 | Derivations, actions of Lie algebras |
| 16N60 | Prime and semiprime associative rings |
| 16W20 | Automorphisms and endomorphisms |
| 47B47 | Commutators, derivations, elementary operators, etc. |
Keywords:
prime ring; skew derivation; \(*\)-commuting \((*\)-centralizing) mapping; antiautomorphism; \(\psi\)-commuting \((\psi\)-centralizing) mappingReferences:
| [1] | A. Abbasi, M. R. Mozumder and N. A. Dar, A note on skew Lie product of prime ring with involution, Miskolc Math. Notes 21 (2020), no. 1, 3-18. · Zbl 1463.16076 |
| [2] | M. K. Abu Nawas and R. M. Al-Omary, On ideals and commutativity of prime rings with generalized derivations, Eur. J. Pure Appl. Math. 11 (2018), no. 1, 79-89. · Zbl 1381.16023 |
| [3] | F. Algarni, A. Alali, H. Alnoghashi, N. Rehman and C. Haetinger, A pair of derivations on prime rings with antiautomorphism, Mathematics 11 (2023), no. 15, Paper No. 3337. |
| [4] | H. Alnoghashi, A. Alali, E. Sogutcu, N. Rehman and F. Algarni, Some classification theorems and endomorphisms in new classes, Mathematics 11 (2023), no. 18, Paper No. 3910. |
| [5] | S. Ali and N. A. Dar, On \ast-centralizing mappings in rings with involution, Georgian Math. J. 21 (2014), no. 1, 25-28. · Zbl 1293.16031 |
| [6] | S. Ali, B. Dhara, B. Fahid and M. A. Raza, On semi(prime) rings and algebras with automorphisms and generalized derivations, Bull. Iranian Math. Soc. 45 (2019), no. 6, 1805-1819. · Zbl 1423.16021 |
| [7] | F. A. A. Almahdi, On “On derivations and commutativity of prime rings with involution”, Georgian Math. J. 28 (2021), no. 3, 331-333. · Zbl 1477.16046 |
| [8] | A. Boua and M. Ashraf, Identities related to generalized derivations in prime *-rings, Georgian Math. J. 28 (2021), no. 2, 193-205. · Zbl 1473.16013 |
| [9] | M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), no. 2, 385-394. · Zbl 0773.16017 |
| [10] | M. Brešar, Commuting maps: A survey, Taiwanese J. Math. 8 (2004), no. 3, 361-397. · Zbl 1078.16032 |
| [11] | M. Chacron, Generalized power commuting antiautomorphisms, Comm. Algebra 49 (2021), no. 11, 5017-5026. · Zbl 1501.16017 |
| [12] | C.-L. Chuang, Differential identities with automorphisms and antiautomorphisms. I, J. Algebra 149 (1992), no. 2, 371-404. · Zbl 0773.16007 |
| [13] | C. L. Chuang, Differential identities with automorphisms and antiautomorphisms. II, J. Algebra 160 (1993), no. 1, 130-171. · Zbl 0793.16014 |
| [14] | C.-L. Chuang and T.-K. Lee, Identities with a single skew derivation, J. Algebra 288 (2005), no. 1, 59-77. · Zbl 1073.16021 |
| [15] | N. J. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III 49 (1955), 19-22. · Zbl 0067.01401 |
| [16] | V. K. Harčenko, Generalized identities with automorphisms, Algebra i Logika 14 (1975), no. 2, 215-237; translation in Algebra and Logic 14 (1975), no. 2, 132-148. · Zbl 0329.16009 |
| [17] | A. Khan, M. Raza and H. Alhazmi, Strong commutativity preserving skew derivations on Banach algebras, J. Taibah Univ. Sci 13 (2019), no. 1, 478-480. |
| [18] | V. K. Kharchenko and A. Z. Popov, Skew derivations of prime rings, Comm. Algebra 20 (1992), no. 11, 3321-3345. · Zbl 0783.16012 |
| [19] | J. H. Mayne, Centralizing automorphisms of Lie ideals in prime rings, Canad. Math. Bull. 35 (1992), no. 4, 510-514. · Zbl 0784.16023 |
| [20] | B. Nejjar, A. Kacha, A. Mamouni and L. Oukhtite, Commutativity theorems in rings with involution, Comm. Algebra 45 (2017), no. 2, 698-708. · Zbl 1370.16036 |
| [21] | L. Oukhtite and O. Ait Zemzami, A study of differential prime rings with involution, Georgian Math. J. 28 (2021), no. 1, 133-139. · Zbl 1469.16043 |
| [22] | E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. · Zbl 0082.03003 |
| [23] | N. u. Rehman and H. M. Alnoghashi, Commutativity of prime rings with generalized derivations and anti-automorphisms, Georgian Math. J. 29 (2022), no. 4, 583-594. · Zbl 1509.16042 |
| [24] | L. Taoufiq and A. Boua, Results in prime rings with two sided α-derivation, Electron. J. Math. Anal. Appl. 4 (2016), no. 2, 38-45. · Zbl 1397.16015 |
| [25] | J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), no. 1, 47-52. · Zbl 0697.16035 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
