Some generalizations of preprojective algebras and their properties

In this note we consider a notion of relative Frobenius pairs of commutative rings S/RS/R. To such a pair, we associate an NN-graded R  -algebra ΠR(S)ΠR(S) which has a simple description and coincides with the preprojective algebra of a quiver with a single central node and several outgoing edges in the split case. If the rank of S over R is 4 and R   is Noetherian, we prove that ΠR(S)ΠR(S) is itself Noetherian and finite over its center and that each ΠR(S)dΠR(S)d is finitely generated projective. We also prove that ΠR(S)ΠR(S) is of finite global dimension if R and S are regular.