  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.
