Outer inverses, minus partial orders, and triplet invariance

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 $a, b, c\in R$, then the triplet $b\hat{a}c$ is invariant under all outer inverses $\hat{a}$ of $a$ if and only if $\operatorname{E}[a]\operatorname{E}[b]\operatorname{E}[c]=0$. Here, for $x\in R$, $\operatorname{E}[x]$ is the smallest idempotent in the extended centroid of $R$ such that $x=\operatorname{E}[x]x$. These answer two questions in Hartwig and Patrício [12].