Integral hyperring - Knowlpedia

Definition

An integral hyperring, also called a hyperdomain, is a commutative hyperring $R$ with $0\ne1$ and no multiplicative zero divisors:

ab=0\quad\Longrightarrow\quad a=0\ \text{or}\ b=0.

This condition concerns the single-valued multiplication. Hyperaddition may still be genuinely multivalued.

Relation to familiar objects

Every hyperfield is an integral hyperring because every nonzero element is multiplicatively invertible. An ordinary commutative ring, regarded as a singleton-addition hyperring, is integral in this sense exactly when it is an integral domain.

An integral hyperring need not be a hyperfield: nonzero elements are required to multiply without producing zero, but they need not all be units.

Role in partial hyperfields

A partial hyperfield selects a multiplicative subgroup of the units of an integral hyperring. The no-zero-divisors condition ensures that the selected nonzero coefficients retain field-like multiplicative behavior even though only part of the ambient hyperaddition is visible.

References

Matthew Baker and Nathan Bowler, “Matroids over partial hyperstructures,” Advances in Mathematics 343 (2019), 821–863. arXiv:1709.09707. Relevant: §2.6, integral hyperrings in the definition of partial hyperfields.
Jaiung Jun, “Algebraic Geometry Over Hyperrings,” Advances in Mathematics 323 (2018), 142–192. arXiv:1512.04837. Relevant: integral hyperrings and their spectra.