In the category of C*-algebras there are many purely algebraic properties which often make life easier, such as the fact that every C*-algebra is idempotent (by Cohen-Hewitt). When speaking of closed ideals, this has a very nice consequence: the intersection of two ideals equals their product.
Motivated by the study of Leavitt path algebras, many C*-algebraic results have been extended to various purely algebraic situations, most of which involve algebras sharing some form of the above properties. It would therefore be interesting to identify a large class of algebras over a given field $K$ to be considered as the natural environment where one would look for generalizations of C*-algebraic results.
To be precise, here is a specific question:
Find a large sub-category of the category of all algebras over a field $K$, including all C*-algebras in case $K=\mathbb C$, in which every algebra is idempotent. The kernels of the morphisms in this category should moreover satisfy the above property according to which intersections and products coincide.
Some people might think of von Neumann regular rings, but not all C*-algebras have this property.
An answer based on first principles would be much more appreciated than any ad hoc description of a category.