Non-commutative Modal Rings and Internalized Equality

We generalize the notion of modal Boolean rings to rings in general. The connection between Boolean rings with equality and modal Boolean rings provides a cue for the definition. The main motivation lies in the existence of examples such as matrix and polynomial rings over modal Boolean rings and Cartesian products of associative rings with identity, along with the desire that the class of "modal rings" be closed under formation of not only the usual homomorphic image, subalgebra and direct product constructions, but also the ring-theoretic constructions of forming matrices and polynomials. © Inst. Math. CAS 2002.