ABSTRACT
Let โ be a prime ring of characteristic different from 2, ๐ฌr be its right Martindale quotient ring, ๐ฌ be its two-sided Martindale quotient ring and ๐ be its extended centroid. Suppose that โฑ, ๐ข are additive mappings from โ into itself and that is a non-central multilinear polynomial over ๐ with n non-commuting variables. We prove the following results:
(a) If โฑ and ๐ข are generalized derivations of โ such that
(a) there exists qโ๐ฌ such that โฑ(x) = xq and ๐ข(x) = qx for all xโโ.
(b) there exist c,qโ๐ฌ such that โฑ(x) = qx+xc, ๐ข(x) = cx+xq for all xโโ, and is central-valued on โ.
(b) If โฑ is a generalized skew derivation of โ such that
for all , then one of the following holds:
(a) there exists ฮปโ๐ such that โฑ(x) = ฮปx for all xโโ;
(b) there exist qโ๐ฌr and ฮปโ๐ such that โฑ(x) = (q+ฮป)x+xq for all xโโ, and is central-valued on โ.
Acknowledgment
This research was done when the first and second authors visited the School of Mathematics and Statistics at Beijing Institute of Technology in the spring of 2013. They take this opportunity to express their sincere thanks to the School of Mathematics and the Office of International Affairs at Beijing Institute of Technology for the hospitality extended to them during their visit. All authors are deeply grateful to a special Training Program of International Exchange and Cooperation of the Beijing Institute of Technology.