On groupoid graded rings (Q1906626)

Let \({\mathcal K}\) be a class of rings which is closed under homomorphic images, ring extensions, one-sided ideals and contains all rings with zero multiplication. Rings \(R=\bigoplus_{s\in S} R_s\) graded by finite groupoids \(S\) are considered. It is shown that the condition: for all finite groupoids \(S\) and all \(S\)-graded rings \(R\), \(R\) belongs to \({\mathcal K}\) if and only if every component \(R_e\), \(e=e^2\in S\), is in \({\mathcal K}\), is equivalent to the respective condition with `groupoids' replaced by `semigroups'. If \({\mathcal K}\) is also on finite sums of one-sided ideals, then these conditions are equivalent to the following assertion: for every finite group \(G\) a \(G\)-graded ring \(R\) is in \({\mathcal K}\) if and only if \(R_e\) is in \({\mathcal K}\), where \(e=e^2\in G\).