Residue structures

We consider residue structures $R/G$ where $(G,+)$ is  an additive subgroup of a   ring $(R,+,\cdot)$, not necessarily an ideal.   Special instances include Krasner's construction of quotient hyperfields, and Dickson's construction of nonassociative algebras. The residue construction, treated formally, satisfies the Noether isomorphism theorems, and may be cast in a setting   which includes direct products and tensor products.