Logics that are generated by idempotents

The main result of this paper is the generalization of the theorem which represents one of the generally accepted cases concerning the characterization of the logic of idempotents (see [5], [6], [7]). If $R$ is a ring then the $R$-circulant matrices are introduced and some consequences for the logics of idempotents of the corresponding rings. Some convenient examples are added as well. Certain results of this paper may find applications in the foundation of quantum theory.