Leavitt path algebras having unbounded generating number
The authors consider the following sequence of ring theoretic properties: {\parindent=0.7cm\begin{itemize}\item[(1)] Invariant basis number (IBN)\item[(2)] Unbounded generating number (UGN), that is, for each positive integer \(m\), any set of generators for the free right \(R\)-module \(R^m\) has cardinality \(\geq m\). \item[(3)] Stable finiteness, that is, for any \(n\in\mathbb N^+\), \(R^n\cong R^n\oplus K\) (as right R-modules) implies \(K=0\). \item[(4)] Hermite property, that is, for all \(m,n\in\mathbb N\) and any right \(R\)-module \(K\), \(R^n\cong R^m\oplus K\) (as right \(R\)-modules) implies that \(n\geq m\) and \(K=R\). \item[(5)] Cancellation of projectives: for any finitely generated projective right \(R\)-modules \(P\) and \(P'\), \(P\oplus R\cong P'\oplus R\) (as right \(R\)-modules) implies that \(P\cong P'\). \end{itemize}} The paper under review studies some of the notions above in the ambient space of Leavitt path algebras (\(L(E)\) in the sequel) and also in that of Cohn path algebras (denoted \(C(E)\) in the sequel). Recently, the first author and \textit{M. Kanuni} [Commun. Algebra 44, No. 1, 371--380 (2016; Zbl 1354.16032)] have shown that \(C(E)\) has IBN for every finite graph \(E\). The authors point out that, at of the writing of this article, it is an open question to give necessary and sufficient conditions on \(E\) so that \(L(E)\) has IBN. However, regarding the remaining four aforementioned properties, they completely classify those graphs \(E\) for which \(L(E)\) has UGN (Theorem 3.16), as well as classify those graphs \(E\) for which \(L(E)\) satisfies (equivalently) properties (3), (4), and (5) (Theorem 4.2). They achieve similar results for \(C(E)\) as well.
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