Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019
Mohammad Ashraf [a] and Bilal Ahmad Wani [a]
On derivations of rings and Banach algebras involving anti-commutator
Pages: 85-97
Received: 3 December 2018
Accepted in revised form: 18 March 2019
Mathematics Subject Classification (2010): 46J10, 16N20, 16N60, 16W25.
Keywords: Semiprime ring, derivation, maximal right ring of quotient, Banach algebra.
Authors address:
[a]: Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
Full Text (PDF)
Abstract:
In the present paper we obtain commutativity of a semiprime ring \( \ \mathcal{R} \ \)
which admits a derivation \(d\) such that either
\( \ (d(x^m \circ y^n))^\ell \pm (x^p \circ_k y^q) = 0 \ \)
for all \( \ x,y\in \mathcal{R} \ \) or \( \ (d(x^m) \circ d(y^n))^\ell \pm (x^p \circ_k y^q) = 0 \ \)
for all \( \ x,y\in \mathcal{R} \ \), where \( \ m,n,p,q,k,\ell \ \) are fixed positive integers.
Finally, we apply the above purely ring theoretic results to Banach algebras and obtain
a noncommutative version of the Singer-Wermer theorem. In particular,
we prove that if \( \ \mathfrak{B} \ \) is a noncommutative Banach algebra which admits a
continuous linear derivation \(d:\mathfrak{B}\rightarrow \mathfrak{B}\) such that
either \( \ (d(x^m \circ y^n))^\ell \pm (x^p \circ_k y^q)\in rad(\mathfrak{B}) \ \)
or \( \ (d(x^m) \circ d(y^n))^\ell \pm (x^p \circ_k y^q)\in rad(\mathfrak{B}) \ \)
for all \( \ x,y\in \mathfrak{B} \ \), where \( \ m,n,p,q,k,\ell \ \) are fixed positive integers,
then \( \ d(\mathfrak{B})\subseteq rad(\mathfrak{B}) \ \).
References
- [1]
-
M. Ashraf and N. Rehman,
On commutativity of rings with derivations, Results Math. 42 (2002), 3-8.
MR1934218
- [2]
-
M. Ashraf, N. Rehman and M. A. Raza,
A note on commutativity of semiprime Banach algebras,
Beitr. Algebra Geom. 57 (2016), 553-560.
MR3535065
- [3]
-
N. Argaç and H. G. Inceboz,
Derivations of prime and semiprime rings,
J. Korean Math. Soc. 46 (2009), 997-1005.
MR2549787
- [4]
-
K. I. Beidar, W. S. Martindale III and A. V. Mikhalev,
Rings with generalized identities,
Monogr. Textbooks Pure Appl. Math., 196, Marcel Dekker, New York, 1996.
MR1368853
- [5]
-
H. E. Bell,
On some commutativity theorems of Herstein,
Arch. Math. (Basel) 24 (1973), 34-38.
MR0320090
- [6]
-
H. E. Bell and M. N. Daif,
On derivations and commutativity in prime rings,
Acta Math. Hungar. 66 (1995), 337-343.
MR1314011
- [7]
-
H. E. Bell and M. N. Daif,
On commutativity and strong commutativity-preserving maps,
Canad. Math. Bull. 37 (1994), 443-447.
MR1303669
- [8]
-
F. F. Bonsall and J. Duncan,
Complete Normed Algebras,
Springer-Verlag, New York-Heidelberg, 1973.
MR0423029
- [9]
-
M. Brešar,
Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings,
Trans. Amer. Math. Soc. 335 (1993), 525-546.
MR1069746
- [10]
-
C. L. Chuang,
GPIs having coefficients in Utumi quotient rings,
Proc. Amer. Math. Soc. 103 (1988), 723-728.
MR0947646
- [11]
-
C. L. Chuang,
Hypercentral derivations, J. Algebra 166 (1994), 39-71.
MR1276816
- [12]
-
M. N. Daif and H. E. Bell,
Remarks on derivations on semiprime rings,
Internat. J. Math. Math. Sci. 15 (1992), 205-206.
MR1143947
- [13]
-
B. Dhara and R. K. Sharma,
Vanishing powers values of commutators with derivations,
Sib. Math. J. 50 (2009), 60-65.
MR2502875
- [14]
-
T. S. Erickson, W. S. Martindale III and J. M. Osborn,
Prime nonassociative algebras,
Pacific J. Math. 60 (1975), 49-63.
MR0382379
- [15]
-
I. N. Herstein,
Topics in ring theory, Univ. of Chicago Press, Chicago-London, 1969.
MR0271135
- [16]
-
I. N. Herstein,
Center-like elements in prime rings,
J. Algebra 60 (1979), 567-574.
MR0549949
- [17]
-
N. Jacobson,
Structure of rings, Colloquium Publications, 37,
Amer. Math. Soc., Provindence, RI, 1956.
MR0081264
- [18]
-
B. E. Johnson and A. M. Sinclair,
Continuity of derivations and a problem of Kaplansky,
Amer. J. Math. 90 (1968), 1067-1073.
MR0239419
- [19]
-
V. K. Kharchenko,
Differential identities of prime rings, Algebra Logic 17 (1979), 155-168.
zbMATH
- [20]
-
B. D. Kim,
On the derivations of semiprime rings and noncommutative Banach algebras,
Acta Math. Sin. (Engl. Ser.) 16 (2000), 21-28.
MR1760520
- [21]
-
C. Lanski,
An Engel condition with derivation for left ideals,
Proc. Amer. Math. Soc. 125 (1997), 339-345.
MR1363174
- [22]
-
T. K. Lee,
Semiprime rings with differential identities,
Bull. Inst. Math. Acad. Sinica 20 (1992), 27-38.
MR1166215
- [23]
-
T. K. Lee,
Semiprime rings with hypercentral derivations,
Canad. Math. Bull. 38 (1995), 445-449.
MR1360594
- [24]
-
W. S. Martindale III,
Prime rings satisfying a generalized polynomial identity,
J. Algebra 12 (1969), 576-584.
MR0238897
- [25]
-
J. H. Mayne,
Centralizing mappings of prime rings,
Canad. Math. Bull. 27 (1984), 122-126.
MR0725261
- [26]
-
K.-H. Park,
On derivations in noncommutative semiprime rings and Banach algebras,
Bull. Korean Math. Soc. 42 (2005), 671-678.
MR2181155
- [27]
-
E. C. Posner,
Derivation in prime rings,
Proc. Amer. Math. Soc. 8 (1957), 1093-1100.
MR0095863
- [28]
-
A. M. Sinclair,
Continuous derivations on Banach algebras,
Proc. Amer. Math. Soc. 20 (1969), 166-170.
MR0233207
- [29]
-
I. M. Singer and J. Wermer,
Derivations on commutative normed algebras,
Math. Ann. 129 (1955), 260-264.
MR0070061
- [30]
-
M. P. Thomas,
The image of a derivation is contained in the radical,
Ann. of Math. 128 (1988), 435-460.
MR0970607
- [31]
-
J. Vukman,
A result concerning derivations in noncommutative Banach algebras,
Glas. Mat. Ser. III 26 (1991), 83-88.
MR1269177
- [32]
-
J. Vukman,
On derivations in prime rings and Banach algebras,
Proc. Amer. Math. Soc. 116 (1992), 877-884.
MR1072093
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