For a free partial action of a group in a set we realize the associated partial skew group ring as an algebra of functions with finite support over an equivalence relation and we use this result to characterize the ideals in the partial skew group ring. This generalizes, to the purely algebraic setting, the known characterization of partial $C^*$-crossed products as groupoid $C^*$-algebras. For completeness we include a new proof of the $C^*$ result for free partial actions.
"Partial crossed products as equivalence relation algebras." Rocky Mountain J. Math. 46 (1) 85 - 104, 2016. https://doi.org/10.1216/RMJ-2016-46-1-85