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 Pumpluen's construction of nonassociative algebras. The residue construction, treated formally, satisfies the Noether isomorphism theorems, and also is cast in a broader categorical setting which includes categorical products, sums, and tensor products.