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Rings in which derivations satisfy certain algebraic conditions. (English) Zbl 0705.16021
The purpose of this paper is to investigate some commutator conditions for rings. Rings in which all inner derivations are potent are called PD- rings. Following are some of the results proved: (1) Let R be a semiprime ring. If d is a derivation of R which is either an endomorphism or an anti-endomorphism, then \(d=0\). (2) Let R be a prime ring and U a nonzero right ideal of R. If d is a derivation of R which acts as a homomorphism or an anti-homomorphism on U, then \(d=0\) on R. (3) Every PD-ring is commutative.
Reviewer: S.K.Jain

16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16N60 Prime and semiprime associative rings
16W25 Derivations, actions of Lie algebras
Full Text: DOI
[1] H. E. Bell, Some commutativity results for periodic rings,Acta Math. Acad. Sci. Hungar.,28 (1976), 279–283. · Zbl 0335.16035 · doi:10.1007/BF01896791
[2] H. E. Bell, On commutativity of periodic rings and near-rings,Acta Math. Acad. Sci. Hungar.,36 (1980), 293–302. · Zbl 0464.16026 · doi:10.1007/BF01898145
[3] A. Giambruno and I. N. Herstein, Derivations with nilpotent values,Rend. Circ. Mat. Palermo, (2)30 (1981), 199–206. · Zbl 0482.16030 · doi:10.1007/BF02844306
[4] N. D. Gupta, Some group laws equivalent to the commutative law,Arch. Math.,17 (1966), 97–102. · Zbl 0135.04302 · doi:10.1007/BF01899854
[5] I. N. Herstein, A theorem on rings,Canad. J. Math.,5 (1953), 238–241. · Zbl 0051.02502 · doi:10.4153/CJM-1953-025-6
[6] I. N. Herstein,Topics in ring theory, Univ. of Chicago Math. Lecture Notes, 1965. · Zbl 0138.26802
[7] A. Kurosh,Theory of groups, v. I, Chelsea Publishing, 1960. · Zbl 0094.24501
[8] F. W. Levi, Groups in which the commutator operation satisfies certain algebraic conditions,J. Indian Math. Soc.,6 (1942), 87–97. · Zbl 0061.02606
[9] F. W. Levi, Notes on group theory I, II,J. Indian Math. Soc.,8 (1944), 1–9. · Zbl 0061.02608
[10] I. D. Macdonald, Some group elements defined by commutators,Math. Scientist,4 (1979), 129–131. · Zbl 0414.20031
[11] W. Streb, Über einen Satz von Herstein und Nakayama,Rend. Sem. Mat. Univ. Padova,64 (1981), 159–171. · Zbl 0474.16024
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