Further results on the inverse along an element in semigroups and rings

In this paper, we introduce a new notion in a semigroup $S$ as an extension of Mary's inverse. Let $a,d\in S$. An element $a$ is called left (resp. right) invertible along $d$ if there exists $b\in S$ such that $bad=d$ (resp. $dab=b$) and $b\leq_\mathcal{L}d$ (resp. $b\leq_\mathcal{R}d$). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) $\pi$-regularity and left (right) $*$-regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally we study the (left, right) inverse along a product in a ring, and, as an application, Mary's inverse along a matrix is expressed.

NF-kB pathway controls mitochondrial dynamics

The Optic atrophy 1 protein (OPA1) is a key element in the dynamics and morphology of mitochondria. We demonstrated that the absence of I?B kinase-a, which is a key element of the nonclassical NF-?B pathway, has an impact on the mitochondrial network morphology and OPA1 expression. In contrast, the absence of NF-?B essential modulator (NEMO) or I?B kinase-ß, both of which are essential for the canonical NF-?B pathway, has no impact on mitochondrial dynamics. Whereas Parkin has been reported to positively regulate the expression of OPA1 through NEMO, herein we found that PARK2 overexpression did not modify the expression of OPA1. PARK2 expression reduced the levels of Bax, and it prevented stress-induced cell death only in Bak-deficient mouse embryonic fibroblast cells. Collectively, our results point out a role of the nonclassical NF-?B pathway in the regulation of mitochondrial dynamics and OPA1 expression.