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ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS

Year 2019, , 87 - 103, 08.01.2019
https://doi.org/10.24330/ieja.504124

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.

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.
Year 2019, , 87 - 103, 08.01.2019
https://doi.org/10.24330/ieja.504124

Abstract

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.
There are 28 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Gurninder S. Sandhu

Deepak Kumar This is me

Publication Date January 8, 2019
Published in Issue Year 2019

Cite

APA Sandhu, G. S., & Kumar, D. (2019). ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS. International Electronic Journal of Algebra, 25(25), 87-103. https://doi.org/10.24330/ieja.504124
AMA Sandhu GS, Kumar D. ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS. IEJA. January 2019;25(25):87-103. doi:10.24330/ieja.504124
Chicago Sandhu, Gurninder S., and Deepak Kumar. “ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS”. International Electronic Journal of Algebra 25, no. 25 (January 2019): 87-103. https://doi.org/10.24330/ieja.504124.
EndNote Sandhu GS, Kumar D (January 1, 2019) ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS. International Electronic Journal of Algebra 25 25 87–103.
IEEE G. S. Sandhu and D. Kumar, “ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS”, IEJA, vol. 25, no. 25, pp. 87–103, 2019, doi: 10.24330/ieja.504124.
ISNAD Sandhu, Gurninder S. - Kumar, Deepak. “ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS”. International Electronic Journal of Algebra 25/25 (January 2019), 87-103. https://doi.org/10.24330/ieja.504124.
JAMA Sandhu GS, Kumar D. ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS. IEJA. 2019;25:87–103.
MLA Sandhu, Gurninder S. and Deepak Kumar. “ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS”. International Electronic Journal of Algebra, vol. 25, no. 25, 2019, pp. 87-103, doi:10.24330/ieja.504124.
Vancouver Sandhu GS, Kumar D. ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS. IEJA. 2019;25(25):87-103.