Hyperring - Knowlpedia

Definition

A hyperring in the Krasner convention is a set $R$ such that $(R,\boxplus,0)$ is a canonical hypergroup, $(R,\cdot,1)$ is a commutative monoid, $0r=0$ , and multiplication distributes over hyperaddition as an equality of subsets:

a(b\boxplus c)=ab\boxplus ac.

Here $aA=\{ax:x\in A\}$ . Multiplication is single-valued; only addition is allowed to be multivalued.

Rings as hyperrings

Every commutative ring becomes a hyperring by interpreting $a+b$ as the singleton $\{a+b\}$ . A hyperring whose every hyper-sum is a singleton is therefore exactly a commutative ring.

Equality is part of the convention

Some broader multiring definitions require only

a(b\boxplus c)\subseteq ab\boxplus ac.

That weaker distributivity is not the house convention. Krasner hyperrings here use equality. This condition concerns the operations inside one hyperring and should not be confused with the inclusion used to define a weak homomorphism between hyperrings.

Incomparability with semirings

Hyperrings generalize rings by making addition multivalued while retaining additive inverses in the canonical-hypergroup sense. Semirings generalize rings by dropping additive inverses while keeping addition single-valued. Consequently neither class contains the other.

References

Alain Connes and Caterina Consani, “The hyperring of adèle classes,” Journal of Number Theory 131 (2011), 159–194. arXiv:1001.4260. Relevant: §2, Krasner hyperrings and quotient hyperrings.
Jaiung Jun, “Algebraic Geometry Over Hyperrings,” Advances in Mathematics 323 (2018), 142–192. arXiv:1512.04837. Relevant: §2, hyperrings and morphisms.