Over algebraically closed fields of arbitrary characteristic, we prove a general multiplicity-freeness theorem for finitely-generated self-projective modules with commutative endomorphism rings. For representations of finite and compact groups, this gives a generalization of Gelfand's criterion on the commutativity of Hecke algebras for Gelfand pairs and multiplicityfree triples. In particular, finite and compact multiplicity-free triples over the complex numbers are also multiplicity-free triples over the algebraic closure of every finite field. Applications include the uniqueness of Whittaker models of Gelfand-Graev representations in equal characteristic and the uniqueness of modular trilinear forms on irreducible representations of quaternion division algebras over local fields.