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No abstract

Let R be a prime ring which is not commutative, with maximal symmetric ring of quotients Q ms R , and let be an anti-automorphism of R. An additive mapwhere a ∈ Q ms R . It is proved that any Jordan -derivation of R is X-inner if either R is not a GPI-ring or R is a PI-ring except when charR = 2 and dim C RC = 4, where C is the extended centroid of R.

In the paper, we obtain an explicit formula for the outer inverses of a regular element in an arbitrary ring. It becomes calculable for outer inverses. We characterize the triplet ba − c (resp. ba + c ) invariant under all inner inverses a − (resp. reflexive inverses a + ) of a in a semiprime ring. It is also proved that if R is a regular ring and a, b, c ∈ R, then the triplet bâc is invariant under all outer inversesâ of a if