Let $R$ be a unital ring, let $E$ be a directed graph and recall that the Leavitt path algebra $L_R(E)$ carries a natural $\mathbb{Z}$-gradation. We show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained.

中文翻译：

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强分级 Leavitt 路径代数

让$R$是一个单位环，让$乙$是一个有向图并回忆一下 Leavitt 路径代数${\mathrm{大号}}_R(乙)$带有天然的$\mathbb{Z}$-层次。我们表明${\mathrm{大号}}_R(乙)$是强烈的$\mathbb{Z}$-分级当且仅当$乙$是行有限的，没有汇，并且满足条件（Y）。我们的结果概括了 Clark、Hazrat 和 Rigby 最近的一个结果，证明很短且自包含。