We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizing previous results from the unital case to a non-unital situation.

中文翻译：

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对象单元群形分级环、交叉积和可分性

我们扩展了 Noether 的由有限伽罗瓦域扩展定义的交叉积代数的经典构造，以涵盖可分离（但不一定是有限或正常）域扩展的情况。这使我们自然而然地考虑我们称为对象单位的特定类型的非单位群群分级环。我们确定此类环何时是强分级、交叉乘积、偏斜群团环和扭曲群团环。我们还获得了对象单位群状分级环何时在其主成分上可分离的必要和充分标准，从而将先前的结果从单位情况推广到非单位情况。