Principally-injective Leavitt path algebras over arbitrary graphs

TL;DR

This paper characterizes when Leavitt path algebras over arbitrary graphs are principally-injective, showing they are equivalent to the graph being acyclic and that such algebras are exactly the von Neumann regular ones.

Contribution

It establishes a complete characterization of principally-injective Leavitt path algebras in terms of graph acyclicity and regularity, linking algebraic and graph-theoretic properties.

Findings

Principally-injective Leavitt path algebras correspond to acyclic graphs.

Such algebras are exactly the von Neumann regular Leavitt path algebras.

The equivalence holds over any arbitrary graph and field.

Abstract

A ring R is called right principally-injective if every R-homomorphism from a principal right ideal aR to R (a in R), extends to R, or equivalently if every system of equations xa=b (a, b in R) is solvable in R. In this paper we show that for any arbitrary graph E and for a field K, principally-injective conditions for the Leavitt path algebra LK(E) are equivalent to that the graph E being acyclic. We also show that the principally injective Leavitt path algebras are precisely the von Neumann regular Leavitt path algebras.