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Let R be a semiprime ring, not necessarily with unity, with extended centroid C. For a ∈ R , let T (a) (respectively I (a) , Ref (a)) denote the set of all outer (respectively inner, reflexive) inverses of a in R. In the paper, we study the inclusion properties of T (a) , I (a) and Ref (a). Among other results, we prove that for a , b ∈ R with a von Neumann regular, Ref (a) ⊆ T (b) (respectively Ref (a) ⊆ I (b)) if and only if a = E [ a ] b (respectively I (a) ⊆ I (b)). Here, E [ a ] is the smallest idempotent in C such that E [ a ] a = a. This gives a common generalization of several known results. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring, not necessarily with unity, and a , b ∈ R. Let I (a) (respectively, Ref (a)) denote the set of inner (respectively, reflexive) inverses of a in R. It is proved that if I (a) ∩ I (b) ≠ ∅ , then I (a) ⊆ I (b) if and only if b = a w b = b w a for all w ∈ I (a). As an immediate consequence, if ∅ ≠ I (a) = I (b) , then a = b (see Theorem 7 in [A. Alahmadi, S. K. Jain and A. Leroy, Regular elements determined by generalized inverses, J. Algebra Appl.18(7) (2019) 1950128] for rings with unity). We also give a generalization of Theorem 10 in [A. Alahmadi, S. K. Jain and A. Leroy, Regular elements determined by generalized inverses, J. Algebra Appl.18(7) (2019) 1950128] by proving that if ∅ ≠ Ref (a) ⊆ Ref (b) then a = b. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring with extended centroid C and with maximal left ring of quotients . An additive map is called a Jordan triple derivation if for all . In 1957, Herstein proved that a Jordan triple derivation, which is also a Jordan derivation, of a noncommutative prime ring of characteristic 2, must be a derivation. In 1989, Brešar proved that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation. In the article, we give a complete characterization of Jordan triple derivations of arbitrary semiprime rings. To get such a characterization we first show that, in some sense, an additive map satisfying for all can be realized as a centralizer with only an exceptional case that and R is commutative. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring with center Z(R) and with extended centroid C. Suppose that τ:R→R is an anti-homomorphism such that the image of τ has small centralizer. It is proved that the following are equivalent: (1) for all x∈R; (2) x+xτ∈Z(R) for all x∈R; (3) xxτ∈Z(R) for all x∈R. In this case, there exists an idempotent e∈C such that (1−e)R is a commutative ring and the semiprime ring eR is equipped with an involution , which is induced canonically by τ. These results generalize both the involution case by Chacron [5] and the anti-automorphism case by the author [7]. [ABSTRACT FROM AUTHOR]