Finite-dimensional algebras admit primitive idempotent decompositions

For any orthogonal decomposition $e=u+v$, the corners $uAu,vAv$ are subspaces of $eAe$. If $u,v\mathrm{\ne }0$, both have positive dimension and each has smaller dimension than $eAe$: for example $v\mathrm{\notin }uAu$, since multiplication by $u$ kills $v$ but fixes every element of $uAu$. Also elements of these two corners multiply to zero in both orders.

Start with the family $\{e\}$ if $e\mathrm{\ne }0$, and the empty family otherwise. Split any nonprimitive member into two nonzero orthogonal idempotents. All members remain mutually orthogonal and sum to $e$. A mutually orthogonal family of nonzero idempotents is linearly independent, since multiplying a linear relation by one member isolates its coefficient. Its size is therefore at most ${\dim }_{k}A$, so after finitely many splits no further split is possible. The terminal family is the asserted primitive decomposition. Equivalently step 1.1 gives induction on corner dimension; a sub-idempotent is primitive in the corner exactly when it is primitive in $A$, because its own corner is the same.