  A central extension is a regular epimorphism in a Barr exact category
$\mathscr{C}$ satisfying suitable conditions involving a given Birkhoff
subcategory of $\mathscr{C}$ (joint work with G. M. Kelly, 1994). In this paper
we take $\mathscr{C}$ to be the category of (not-necessarily-unital) algebras
over a (unital) commutative ring and consider central extensions with respect
to the category of commutative algebras. We propose a new approach that avoids
the intermediate notion of central extension due to A. Fr\"ohlich in showing
that $\alpha:A\to B$ is a central extension if and only if $aa'=a'a$ for all
$a,a'\in A$ with $\alpha(a')=0$. This approach motivates introducing what we
call $\textit{weakly action representable categories}$, and we show that such
categories are always action accessible. We also make remarks on what we call
$\textit{initial weak representations of actions}$ and formulate several open
questions.
