Integral path ring of a finite quiver

A finite quiver is a quadruple $Q=(Q_0,Q_1,s,t)$ in which $Q_0$ and $Q_1$ are finite sets, whose elements are called vertices and arrows, together with maps $s,t:Q_1\to Q_0$ assigning to an arrow $a$ its source $s(a)$ and its target $t(a)$; one draws $a:s(a)\to t(a)$. Both sets are finite and no further structure is assumed: $Q$ may have several arrows with the same source and target, and an arrow may be a loop, although the quivers used below have neither property.

A directed path in $Q$ of length $l\ge 0$ consists of vertices $v_0,\dots ,v_l$ and specified arrows $a_j:v_{j-1}\to v_j$ for $1\le j\le l$. Write it as $p=(v_0\stackrel{a_1}{\to }{v}_1\stackrel{a_2}{\to }\cdots \stackrel{a_l}{\to }{v}_l)$. The arrows are part of the data: distinct parallel arrows give distinct paths. A path of length zero is a single vertex $(v)$, called a vertex idempotent. Write $\mathrm{\mid }p\mathrm{\mid }:=l$, $s(p):=v_0$ and $t(p):=v_l$ for its length, source and target. When the arrows are determined by their endpoints, as in the doubled lines used below, abbreviate this by $(v_0\mathrm{\mid }\cdots \mathrm{\mid }{v}_l)$; in formulas for a general quiver this abbreviation retains the specified arrows as implicit data. Paths $p$ and $q$ are composable if $t(p)=s(q)$. Their concatenation $pq$ traverses the arrow sequence of $p$ first and then that of $q$, retaining all arrow labels; in the abbreviated notation it is $(v_0\mathrm{\mid }\cdots \mathrm{\mid }{v}_l)(v_l\mathrm{\mid }{w}_1\mathrm{\mid }\cdots \mathrm{\mid }{w}_{l^{\mathrm{\prime }}})=(v_0\mathrm{\mid }\cdots \mathrm{\mid }{v}_l\mathrm{\mid }{w}_1\mathrm{\mid }\cdots \mathrm{\mid }{w}_{l^{\mathrm{\prime }}})\mathrm{.}$

The integral path ring $\mathbb{Z}Q$ is the free abelian group on the set of all directed paths of $Q$ (The free module on a set and its standard basis), with multiplication defined on basis paths by $(v_0\mathrm{\mid }\cdots \mathrm{\mid }{v}_l)(w_0\mathrm{\mid }\cdots \mathrm{\mid }{w}_{l^{\mathrm{\prime }}}):=\{\begin{array}{ll}(v_0\mathrm{\mid }\cdots \mathrm{\mid }{v}_l\mathrm{\mid }{w}_1\mathrm{\mid }\cdots \mathrm{\mid }{w}_{l^{\mathrm{\prime }}}),& v_l=w_0,\\ 0,& v_l\mathrm{\ne }{w}_0,\end{array}$ and extended by $\mathbb{Z}$-bilinearity. Its unit is the finite sum $1_{\mathbb{Z}Q}={\sum }_{v\in Q_0}(v)$ of the vertex idempotents, which is a finite sum because $Q_0$ is finite.

Endpoint conventions. For a path $p$ and vertices $u,v$ one has $(u)p=\{\begin{array}{ll}p,& s(p)=u,\\ 0,& s(p)\mathrm{\ne }u,\end{array}p(v)=\{\begin{array}{ll}p,& t(p)=v,\\ 0,& t(p)\mathrm{\ne }v\mathrm{.}\end{array}$ Consequently the vertex idempotents are mutually orthogonal, $(u)(v)={\delta }_{uv}(v)$, and they decompose the ring as $\mathbb{Z}Q={⨁}_{u,v\in Q_0}(u)\mathbb{Z}Q(v),$ the sum of the free abelian groups spanned by the paths with source $u$ and target $v$. In the quotient rings used below, the analogous statements read: the right ideal generated by the image of $(u)$ is spanned by the classes of paths with source $u$, and the left ideal generated by the image of $(v)$ is spanned by the classes of paths with target $v$, so that $A{e}_i$ consists of the paths ending at $i$ and $e_{i}A$ of the paths beginning at $i$ whenever $A$ is a quotient of $\mathbb{Z}Q$ and $e_i$ is the image of $(i)$.

Empty quiver. For $Q_0=Q_1=\mathrm{\varnothing }$ the set of paths is empty, so $\mathbb{Z}Q$ is the zero abelian group, the displayed sum for $1$ is empty and $1=0$; this is the zero ring, and every statement below is trivial there. All quivers occurring on this page are the doubled lines ${\mathrm{\Gamma }}_m$ with $m\ge 1$, which have $m+1\ge 2$ vertices.