ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS

Year 2019, Volume: 25 Issue: 25, 87 - 103, 08.01.2019

Gurninder S. Sandhu Deepak Kumar

https://doi.org/10.24330/ieja.504124

Cited By: 1

Abstract

 Let $R$ be a ring, a mapping $F:R\rightarrow R$ together with a mapping $d:R\rightarrow R$

is called a multiplicative (generalized)-reverse derivation if

$F(xy)=F(y)x+yd(x)$ for all $x,y\in R$. The aim of this note is to

investigate the commutativity of prime rings admitting

multiplicative (generalized)-reverse derivations. Precisely, it is

proved that for some nonzero element $a$ in $R$ the conditions:

$a(F(xy)\pm xy)=0$, $a(F(x)F(y)\pm xy)=0$, $a(F(xy)\pm

F(y)F(x))=0$, $a(F(x)F(y)\pm yx)=0$, $a(F(xy)\pm yx)=0$ are

sufficient for the commutativity of $R$. Moreover, we describe the

possible forms of generalized reverse derivations of prime rings.

Keywords

Prime ring, multiplicative (generalized)-reverse derivation, generalized reverse derivation, annihilator conditions

References

• A. Aboubakr and S. Gonzalez, Reverse generalized derivations of semiprime rings, Sib. Math. J., 56(2) (2015), 199-205.
• A. Ali, D. Kumar and P. Miyan, On generalized derivations and commutativity of prime and semiprime rings, Hacet. J. Math. Stat., 40(3) (2011), 367-374.
• S. Ali, B. Dhara, N. A. Dar and A. N. Khan, On Lie ideals with multiplicative (generalized)-derivations in prime and semiprime rings, Beitr. Algebra Geom., 56(1) (2015), 325-337.
• M. Ashraf, A. Ali and S. Ali, Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math., 31(3) (2007), 415-421.
• M. Ashraf and N. Rehman, On derivation and commutativity in prime rings, East-West J. Math., 3(1) (2001), 87-91.
• H. E. Bell and W. S. Martindale, III, Centralizing mappings of semiprime rings, Canad. Math. Bull., 30(1) (1987), 92-101.
• M. Bresar, On the distance of composition of two derivations to the generalized derivation, Glasgow Math. J., 33(1) (1991), 89-93.
• D. K. Camci and N. Aydin, On multiplicative (generalized)-derivations in semiprime rings, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 66(1) (2017), 153-164.
• C. M. Chang and Y. C. Lin, Derivations on one-sided ideals of prime rings, Tamsui Oxf. J. Math. Sci., 17(2) (2001), 139-145.
• M. N. Daif, When is a multiplicative derivation additive?, Internat. J. Math. Math. Sci., 14(3) (1991), 615-618.
• M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Inter- nat. J. Math. Math. Sci., 15(1) (1992), 205-206.
• M. N. Daif and M. S. Tammam-El-Sayiad, Multiplicative generalized deriva- tions which are additive, East-West J. Math., 9(1) (2007), 31-37.
• B. Dhara and S. Ali, On multiplicative (generalized)-derivations in prime and semiprime rings, Aequationes Math., 86(1-2) (2013), 65-79.
• A. K. Faraj, On generalized (,)-reverse derivations of prime rings, Iraqi J. Sci., 52(2) (2011), 218-224.
• H. Goldmann and P. Semrl, Multiplicative derivations on C(X), Monatsh. Math., 121(3) (1996), 189-197.
• I. Gusic, A note on generalized derivations of prime rings, Glas. Mat. Ser. III, 40(60) (2005), 47-49.
• I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), 1104-1110.
• B. Hvala, Generalized derivations in rings, Comm. Algebra, 26(4) (1998), 1147- 1166.
• J. Pinter-Lucke, Commutativity conditions for rings: 1950-2005, Expo. Math., 25(2) (2007), 165-174.
• W. S. Martindale, III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc., 21 (1969), 695-698.
• H. Marubayashi, M. Ashraf, N. Rehman and S. Ali, On generalized (, )- derivations in prime rings, Algebra Colloq., 17(Spec. 1) (2010), 865-874.
• J. H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull., 27(1) (1984), 122-126.
• E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.
• N. Rehman, On commutativity of rings with generalized derivations, Math. J. Okayama Univ., 44 (2002), 43-49.
• M. Samman and N. Alyamani, Derivations and reverse derivations in semiprime rings, Int. Math. Forum, 2(39) (2007), 1895-1902.
• G. S. Sandhu and D. Kumar, A note on derivations and Jordan ideals of prime rings, AIMS Math., 2(4) (2017), 580-585.
• G. S. Sandhu and D. Kumar, Derivable mappings and commutativity of asso- ciative rings, Italian J. Pure Appl. Math., 40 (2018), 376-393.
• S. K. Tiwari, R. K. Sharma and B. Dhara, Some theorems of commutativity on semiprime rings with mappings, Southeast Asian Bull. Math., 42(2) (2018), 279-292.