Given a partial action $\alpha=(A_g,\alpha_g)_{g\in \mathcal{G}}$ of a connected groupoid $\mathcal{G}$ on a ring $A$ and an object $x$ of $\mathcal{G}$, the isotropy group $\mathcal{G}(x)$ acts partially on the ideal $A_x$ of $A$ by the restriction of $\alpha$. In this paper we investigate the following reverse question: under what conditions a partial group action of $\mathcal{G}(x)$ on an ideal of $A$ can be extended to a partial groupoid action of $\mathcal{G}$ on $A$? The globalization problem and some applications to the Morita and Galois theories are also considered, as extensions of similar results from the group actions case.

中文翻译：

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部分动作的限制和扩展

给定环 $A$ 上的连接群群 $\mathcal{G}$ 的部分动作 $\alpha=(A_g,\alpha_g)_{g\in\mathcal{G}}$ 和对象 $x$ $\mathcal{G}$，各向同性群 $\mathcal{G}(x)$ 通过 $\alpha$ 的限制作用于 $A$ 的理想 $A_x$。在本文中，我们研究了以下相反的问题：在什么条件下，$\mathcal{G}(x)$ 对 $A$ 理想的偏群作用可以扩展为 $\mathcal{G}美元兑美元？全球化问题和 Morita 和 Galois 理论的一些应用也被考虑，作为群体行动案例类似结果的扩展。