|Title:||A note on inner and reflexive inverses in semiprime rings||Authors:||TSIU-KWEN LEE||Keywords:||inner inverse | reflexive inverse | Semiprime ring | von Neumann regular element||Issue Date:||1-Jan-2019||Source:||Journal of Algebra and its Applications||Abstract:||
© 2020 World Scientific Publishing Company. 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) â I(b)â‰ â then I(a) âŠ I(b) if and only if b = awb = bwa for all w I(a). As an immediate consequence, if ââ‰ 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 ââ‰ Ref(a) âŠRef(b) then a = b.
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