Some Extension Theorems in the Ring of Quotients of $$*$$-Prime Rings

Abstract

Let R be a semiprime ring with an involution ‘\(*\)’. Let \(Q_{mr}\) and \(Q_s\) denote its right Utumi quotient ring and right symmetric Martindale quotient ring, respectively. In the present paper, the following extension problems have been obtained: (i) an involution of a semiprime ring can be uniquely extended to its right symmetric Martindale quotient ring; (ii) if R is a \(*\)-prime ring, then so is its right symmetric Martindale quotient ring; (iii) every \(*\)-derivation of a commutative semiprime ring can be uniquely extended to its right symmetric Martindale quotient ring. Finally, we have also discussed C-dependence of any two nonzero elements of right symmetric Martindale quotient ring of \(*\)-prime ring R, where C is the extended centroid of R.