论文标题
局部偏斜型环的理想结构，并应用于拓扑动态和超级代数
The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras
论文作者
论文摘要
鉴于环$ r $的部分操作$α$ groupoid $ g $，我们研究了相关的部分偏度群体环$ r \rtimes_αg $，它带有天然的$ g $ rading。我们表明，$ r $的$ g $ invariant理想与$ g $ rad的戒指的分级理想之间有一对一的对应关系。具有剩余的交点属性。 Furthermore, if $α$ is induced by a topological partial action $θ$, then we prove that minimality of $θ$ is equivalent to $G$-simplicity of $R$, topological transitivity of $θ$ is equivalent to $G$-primeness of $R$, and topological freeness of $θ$ on every closed invariant subset of the underlying topological space is equivalent to $α$具有残差交叉属性。作为一种应用，我们根据相关部分作用的拓扑特性以及相关的Ultragraph代数的代数特性来表征超大的条件（k）。
Given a partial action $α$ of a groupoid $G$ on a ring $R$, we study the associated partial skew groupoid ring $R \rtimes_α G$, which carries a natural $G$-grading. We show that there is a one-to-one correspondence between the $G$-invariant ideals of $R$ and the graded ideals of the $G$-graded ring $R \rtimes_αG.$ We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of $R \rtimes_αG.$ We show that every ideal of $R \rtimes_αG$ is graded if, and only if, $α$ has the residual intersection property. Furthermore, if $α$ is induced by a topological partial action $θ$, then we prove that minimality of $θ$ is equivalent to $G$-simplicity of $R$, topological transitivity of $θ$ is equivalent to $G$-primeness of $R$, and topological freeness of $θ$ on every closed invariant subset of the underlying topological space is equivalent to $α$ having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra.
