Resumen de Prime rings with involution involving left multipliers

Abdelkarim Boua, Muhammad Ashraf

• Let R be a prime ring of characteristic different from 2 with involution ’∗’ of the second kind and n ≥ 1 be a fixed positive integer. In the present paper it is shown that if R admits nonzero left multipliers S and T, then the following conditions are equivalent: (i) R is commutative. (ii) Tn([x, x∗]) 2 Z(R) for all x 2 R; (iii) Tn(x ◦ x∗) 2 Z(R) for all x 2 R; (iv) [S(x), T(x∗)] 2 Z(R) for all x 2 R; (v) [S(x), T(x∗)] - (x ◦ x∗) 2 Z(R) for all x 2 R; (vi) S(x) ◦ T(x∗) 2 Z(R) for all x 2 R; (vii) S(x) ◦ T(x∗) - [x, x∗] 2 Z(R) for all x 2 R. The existence of hypotheses in various theorems have been justified by the examples.