Elementary Magma Gradings on Rings

Suppose that $G$ and $H$ are magmas and that $R$ is a strongly $G$-graded ring. We show that there is a bijection between the set of elementary (nonzero) $H$-gradings of $R$ and the set of (zero) magma homomorphisms from $G$ to $H$. Thereby we generalize a result by D\u{a}sc\u{a}lescu, N\u{a}st\u{a}sescu and Rios Montes from group gradings of matrix rings to strongly magma graded rings. We also show that there is an isomorphism between the preordered set of elementary (nonzero) $H$-filters on $R$ and the preordered set of (zero) submagmas of $G \times H$. These results are applied to category graded rings and, in particular, to the case when $G$ and $H$ are groupoids. In the latter case, we use this bijection to determine the cardinality of the set of elementary $H$-gradings on $R$.