Path algebra

The path algebra $kQ$ has the directed paths in a quiver $Q$ as a basis; multiplication concatenates composable paths and is zero otherwise. Representations of $Q$ are naturally modules over $kQ$.

Closed paths at vertex $i$ form a basis of $e_{i}kQ{e}_i$. 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 $fg\ne 0$. 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.

The vertex projective module of a path algebra $P(i)=kQ{e}_i$ has at vertex $j$ the span of paths from $i$ to $j$. Each arrow appends itself to such paths. It is a projective module because it is a summand of the free module $kQ$. Its endomorphism ring is $(e_{i}kQ{e}_i{)}^{\mathrm{op}}$, acting by right multiplication. The closed-path corner of a path algebra is a domain, so this ring has no nontrivial idempotents and $P(i)$ is indecomposable. Directed cycles may make these projectives infinite-dimensional.

Evaluation at $e_i$ identifies a module homomorphism $h:kQ{e}_i\to X$ with $h(e_i)\in e_{i}X$. Given $x\in e_{i}X$, the inverse sends $p$ to $px$. The isomorphism is natural and shows that Hom functor evaluation on a vertex projective is exact.

For a quiver with finitely many vertices, send a representation of a quiver $(X_i,f_{\rho })$ to ${⨁}_i{X}_i$. Vertex idempotents act as projections, and paths act as composites of the arrow maps. Conversely, a unital module $M$ gives vertex spaces $e_{i}M$ and arrow maps $x\mapsto \rho x$. Compatible vertex maps are exactly module homomorphisms. These constructions give inverse equivalences of categories.

This two-sided ideal is spanned by paths of positive length; its $m$th power is spanned by paths of length at least $m$. The quotient is a product of copies of the ground field indexed by vertices. For a finite acyclic quiver, the ideal is nilpotent and equals the Jacobson radical. A loop shows that the equality can fail in general.