Let G be a group with neutral element e and let \(S=\bigoplus _{g \in G}S_{g}\) be a G-graded ring. A necessary condition for S to be noetherian is that the principal component S_e is noetherian. The following partial converse is well-known: If S is strongly-graded and G is a polycyclic-by-finite group, then S_e being noetherian implies that S is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings. As our main application we obtain characterizations of noetherian and artinian Leavitt path algebras with coefficients in a general unital ring. This extends a recent characterization by Steinberg for Leavitt path algebras with coefficients in a commutative unital ring and previous characterizations by Abrams, Aranda Pino and Siles Molina for Leavitt path algebras with coefficients in a field. Secondly, we obtain characterizations of noetherian and artinian unital partial crossed products.

中文翻译：

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Epsilon强梯度环的链条件及其在Leavitt路径代数中的应用

令G为具有中性元素e的基团，令\（S = \ bigoplus _ {g \ in G} S_ {g} \）为G级环。S为noetherian的必要条件是主成分S _e为noetherian。以下部分逆是众所周知的：如果S是强渐变的，并且G是一个多环有限基团，则S _e为noetherian意味着S是noetherian。我们将noetherianity结果推广到最近推出的一类epsilon强梯度环。我们还将提供有关ε级渐变环的手艺度的结果。作为我们的主要应用，我们获得了在普通unit环中具有系数的Noetherian和Artinian Leavitt路径代数的刻画。这扩展了斯坦伯格最近对具有可交换unit环系数的Leavitt路径代数的刻画，以及先前Abrams，Aranda Pino和Siles Molina对具有一个系数的Leavitt路径代数的先前刻画。其次，我们获得了noetherian和artinian单一部分交叉产物的特征。