Central generalized bi-semi-derivations on semiprime rings

In this research, our goal is to characterize the structure of central generalized bi-semiderivation ? on ring. Infact, we obtain a few commutativity observations for bi-semi-derivations that commute on prime and semiprime ring. A non-commutative version of some results is also investigated with he help of algebraic identities in which ? will acting as left centralizer.

[1] G. Maksa, “A remark on symmetric biadditive functions having non-negative diagonalization,” Glasnik. Mat., vol. 15, no. 35, pp. 279–282, 1980.

[2] J. Bergen, “Derivations in prime rings,” Canad. Math. Bull., vol. 26, no. 8, pp. 267–270, 1983.

[3] A. Ali, D. V., and F. Shujat, “Results concerning symmetric generalized biderivations of prime and semiprime rings,” Matematiqki Vesnik, vol. 66, no. 4, pp. 410–417, 2014.

[4] S. Ali and H. Shuliang, “On derivations in semiprime rings,” Algebr. Represent. Theory, vol. 15, no. 0, pp. 1023–1033, 2012.

[5] J. C. Chang, “On semiderivations of prime rings,” Chinese J. Math., vol. 12, pp. 255–262, 1984.

[6] I. N. Herstein, “A note on derivations ii,” Canad. Math. Bull., vol. 22, pp. 509–511, 1979.

[7] N. Rehman and A. Z. Ansari, “On lie ideals with symmetric bi-additive maps in rings,” Palestine J. Math., vol. 2, pp. 14–21, 2013.

[8] F. Shujat, “On symmetric generalized bi-semiderivations of prime rings,” Bol. Soc. Paran. Mat., vol. 42, pp. 1–5, 2024.

[9] T. Lam, “A first course in noncommutative rings,” Graduate Texts in Mathematics, 2001.

[10] I. N. Herstein, “Rings with involution,” University of Chicago Press, 1976.

[11] F. Shujat, “Additive multipliers and bi-semiderivations on rings,” Ann. Math. Comp. Sci., vol. 4, pp. 1–6, 2021.

[12] B. Zalar, “On centralizers of semiprime rings,” Comment. Math. Univ. Carol., vol. 32, pp. 609–614, 1991.