Quotient hyperring - Knowlpedia

Core idea

Let $R$ be a commutative ring and let $G\le R^\times$ be a multiplicative subgroup of its units. The quotient hyperring $R/G$ is the set of multiplicative orbits

[a]=aG

with multiplication $[a][b]=[ab]$ and hyperaddition

[a]\boxplus[b] =\{[ag+bh]:g,h\in G\}.

These operations are independent of the chosen representatives and make $R/G$ a Krasner hyperring.

Why addition is multivalued

A quotient representative can be rescaled independently before addition. Different choices of $g$ and $h$ can place $ag+bh$ in different orbits, so addition returns the set of all possible resulting orbits. Multiplication is single-valued because rescaling $a$ and $b$ only rescales $ab$ by an element of $G$ .

The quotient map

The orbit map

\pi:R\longrightarrow R/G,\qquad a\longmapsto[a]

is a weak hyperring homomorphism after $R$ is viewed as a singleton-valued hyperring. It is generally not strong: the image $\{[a+b]\}$ can be a proper subset of $[a]\boxplus[b]$ .

Hyperfield case

If $R=K$ is a field, every nonzero orbit has a multiplicative inverse, so $K/G$ is a quotient hyperfield. The Krasner and sign hyperfields arise this way. This multiplicative-orbit construction is not the ordinary quotient ring $R/I$ by an additive ideal.

References

Alain Connes and Caterina Consani, “The hyperring of adèle classes,” Journal of Number Theory 131 (2011), 159–194. arXiv:1001.4260. Relevant: Proposition 2.6 and quotient hyperrings $R/G$ .
Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034. Relevant: factor multirings and factor hyperfields.