Let $Q$ be an infinite quiver without oriented cycle. Is it true that the radical of $KQ$ is generated by all the arrows? Note that we're allowing for infinitely long directed paths. What can we say about its socle?
2026-03-25 19:07:20.1774465640
What's the radical and socle of the path algebra of an infinite acyclic quiver?
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Yes, the radical of $KQ$ is the ideal generated by all the arrows, see e.g. this mathoverflow question and the (more general) reference in the comment.
To answer the question about the socle observe first that for a module $M$, the submodule $\operatorname{soc}(M)$ is the subspace of all elements $m\in M$ such that all arrows act as zero (otherwise $\operatorname{rad}(A)m$ is a non-zero proper submodule of $M$). In particular, for the path algebra $KQ$, its socle is spanned by all paths ending in sinks.