Triangular primitive-idempotent decomposition
= Triangular primitive-idempotent decomposition
{title2=$e_ie_j=0\ (i>j)\Longrightarrow\bigoplus_iAe_i\subseteq A$}
If <idempotents> $e_1,\ldots,e_N$ satisfy $e_ie_j=0$ for $i>j$, then the <left ideals> $Ae_i$ have an internal <direct sum>. Multiply a relation $\sum z_i=0$ by $e_1$ on the right to obtain $z_1=0$, and continue in order. Pairwise orthogonality in both directions is unnecessary. If the sum has dimension $\dim A$, it is the whole algebra.