A ring R is said to be an S-graded ring inducing S if there exists a family \(\{R_s\}_{s\in S}\) of nonzero additive subgroups of R, known as components of R, indexed by a partial groupoid (magma) S, that is, by a set with a partial binary operation, such that \(R=\bigoplus _{s\in S}R_s,\) and: (i) \(R_s R_t\subseteq R_{st}\) whenever st is defined; (ii) \(R_s R_t\ne 0\) if and only if the product st is defined. The class of S-graded rings inducing S includes all the other classes of graded rings. Group graded quasi-Frobenius and group graded Frobenius rings (with unity) are introduced and studied in Dǎscǎlescu et al. (J Algebra 620:392–424, 2023). In this paper, we study graded quasi-Frobenius and graded Frobenius rings in the S-graded rings inducing S setting, under assumptions that rings are with unity and that they are graded by cancellative S. We also examine how the property of R being graded Frobenius depends on the property of each ring component of R being Frobenius.
