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A central extension is a regular epimorphism in a Barr exact category$\mathscr{C}$ satisfying suitable conditions involving a given Birkhoffsubcategory of $\mathscr{C}$ (joint work with G. M. Kelly, 1994). In this paperwe take $\mathscr{C}$ to be the category of (not-necessarily-unital) algebrasover a (unital) commutative ring and consider central extensions with respectto the category of commutative algebras. We propose a new approach that avoidsthe intermediate notion of central extension due to A. Fr\"ohlich in showingthat $\alpha:A\to B$ 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 wecall $\textit{weakly action representable categories}$, and we show that suchcategories are always action accessible. We also make remarks on what we call$\textit{initial weak representations of actions}$ and formulate several openquestions.

Central extensions of associative algebras and weakly action representable categories