Let $A$ be an algebra, say over $\mathbb{C}$ and finite-dimensional, but not necessary semisimple. I have the strong feeling, which I would like to prove and use, about the following rather natural approach.

Consider the set $S$ of central idempotents in $A$, then I want to turn the set of central idempotents $\Delta$ into an **abstract simplicial complex**. Two idempotents $e_1,e_2$ are connected iff $e_1\in e_2A$, simplices consist of completely connected subset and and the highest-dimensional simplices hence are flags associated to a decomposition of $A$ into central **primitive** idempotents. This is much how one defines buildings as simplicial complexes e.g. of subspaces of a finite vector space....

Is this OK? I'm not familiar with calculating with central prinimtive idempotents - especially I'm concerned, **if any decomposistion into central primitive idempotents has the same cardinality??**

..**if yes**,it seems so natural, something similar ought to exist (surely more advanced ;-) - **any reference?** I would probably also suffice with a related construction and/or I want to find out more established facts about such an "algebraic simplicial complex / building"...Thanks for your help in advance.