Narrow Results

Let $R$ be a semiprime ring, not necessarily with unity, and $a,b\in R$. Let $I(a)$ (respectively, $\mathrm{\text{Ref}}(a)$) denote the set of inner (respectively, reflexive) inverses of $a$ in $R$. It is proved that if $I(a)\cap I(b)\ne \varnothing$, then $I(a)\subseteq I(b)$ if and only if $b=awb=bwa$ for all $w\in I(a)$. As an immediate consequence, if $\varnothing \ne I(a)=I(b)$, 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 $\varnothing \ne \mathrm{\text{Ref}}(a)\subseteq \mathrm{\text{Ref}}(b)$ then $a=b$.

We give an example to show that, for nonunital rings $R$, the direct sum $aR\oplus bR=(a+b)R$ with $a+b$ regular has no in general right-left symmetry. It is then proved that the right-left symmetry actually holds in a left and right faithful ring.