Prime rings with involution involving left multipliers

Abstract

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∗]) ∈ Z(R) for all x ∈ R; (iii) Tn(x ◦ x∗) ∈ Z(R) for all x ∈ R; (iv) [S(x), T (x∗)] ∈ Z(R) for all x ∈ R; (v) [S(x), T (x∗)] − (x ◦ x∗) ∈ Z(R) for all x ∈ R; (vi) S(x) ◦ T (x∗) ∈ Z(R) for all x ∈ R; (vii) S(x) ◦ T (x∗) − [x, x∗] ∈ Z(R) for all x ∈ R. The existence of hypotheses in various theorems have been justified by the examples.