= Closed-path corner of a path algebra is a domain
{title2=$e_ikQe_i$}
Closed paths at vertex $i$ form a <basis> of $e_ikQe_i$. For nonzero finite linear combinations $f,g$, take their largest path lengths $a,b$. In the length-$a+b$ part of $fg$, a concatenated path has a unique cut into lengths $a,b$. Thus a product of nonzero top-degree coefficients cannot cancel, and $fg\ne0$. The corner is a <noncommutative domain> and has no <idempotents> except zero and its identity $e_i$. This argument works with oriented cycles and proves indecomposability of the <vertex projective module of a path algebra> without falsely assuming that its endomorphism ring is local.
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