On Jordan ∗-mappings in rings with involution

• Main Authors: Shakir Ali, Nadeem Ahmad Dar, Dušan Pagon
• Format: Article
• Language: English
• Published: SpringerOpen 2016-01-01
• Series: Journal of the Egyptian Mathematical Society
• Subjects: Prime ring; Involution; Jordan ∗-derivation; Jordan left ∗-derivation; Symmetric Jordan ∗-biderivation; Symmetric Jordan triple ∗-biderivation.
• Online Access: http://www.sciencedirect.com/science/article/pii/S1110256X14001370

The objective of this paper is to study Jordan ∗-mappings in rings with involution ∗. In particular, we prove that if R is a prime ring with involution ∗, of characteristic different from 2 and D is a nonzero Jordan ∗-derivation of R such that [D(x),x]=0, for all x∈R and S(R)∩Z(R)≠(0), then R is commutative. Further, we also prove a similar result in the setting of Jordan left ∗-derivation. Finally, we prove that any symmetric Jordan triple ∗-biderivation on a 2-torsion free semiprime ring with involution ∗ is a symmetric Jordan ∗-biderivation.