Some commutativity theorems for $*$-prime rings with $(\sigma,\tau)$-derivation

Document Type : Research Paper


1 Department of Mathematics,‎ ‎Aligarh Muslim University‎, ‎Aligarh‎, ‎202002, India.

2 Department of Mathematics,‎ ‎Aligarh Muslim University‎, ‎Aligarh‎, ‎202002, ‎India.


‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(\sigma,\tau)$-derivation of $R$ with associated‎ ‎automorphisms $\sigma$ and $\tau$ of $R$‎, ‎such that $\sigma$‎, ‎$\tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{\sigma,\tau}=\{c\in‎ ‎R~|~c\sigma(x)=\tau(x)c~\mbox{for~all}~x\in R\}.$ In the present paper‎, ‎it is shown that if characteristic of $R$ is different from two and‎ ‎$[d(U),d(U)]_{\sigma,\tau}=\{0\},$ then $R$ is commutative‎. ‎Commutativity of $R$ has also been established in case if‎ ‎$[d(R),d(R)]_{\sigma,\tau}\subseteq C_{\sigma,\tau}.$


Main Subjects

M. Ashraf and A. Khan, Commutativity of  prime rings with generalized derivations, Rend. Semin. Mat. Univ. Padova 125 (2011) 75--79.
M. Ashraf and N. Rehman, On derivation and commutativity in prime rings, East-West J. Math. 3 (2001), no. 1, 87--91.
M. Ashraf and N. Rehman, On commutativity of rings with derivation, Results Math. 42 (2002), no. 1-2, 3--8.
N. Aydin and K. Kaya, Some generalizations in prime rings with $(σ,τ)$-derivation, Doga Mat. 16 (1992), no. 3, 169--176.
I. N. Herstein, Rings with Involution, University of Chicago press, Chicago, 1976.
I. N. Herstein, A note on derivations, Canad. Math. Bull. 21 (1978), no. 3, 369--370.
I. N. Herstein, A note on derivations II, Canad. Math. Bull. 22 (1979), no. 4, 509--511.
S. Huang, Some generalizations in certain classes of rings with involution, Bol. Soc. Paran. Mat. (3) 29 (2011), no. 1, 9--16.
K. Kaya, On  $(σ,τ)$-derivations of prime rings, (Turkish) Doga Mat. 12 (1988), no. 2, 42--45.
P. H. Lee and T. K. Lee, On derivations of prime rings, Chin. J. Math. 9 (1981), no. 2, 107--110.
L. Okhtite, On derivations in prime rings, Int. J. Algebra 1 (2007), no. 5-8, 241--246.
L. Okhtite, Some properties of derivations on rings with involution, Int. J. Mod. Math. 4 (2009), no. 3, 309--315.
L. Okhtite, Commutativity conditions on derivations and Lie ideals in prime rings, Beitr. Algebra Geom. 51 (2010), no. 1, 275--282.