  The article targets binomial ideals in quantum tori and quantum affine
spaces. First, noncommutative analogs of known results for commutative
(Laurent) polynomial rings are obtained, including the following: Under the
assumption of an algebraically closed base field, it is proved that primitive
ideals are binomial, as are radicals of binomial ideals and prime ideals
minimal over binomial ideals. In the case of a quantum torus
$\mathcal{T}_{\bf{q}}$, the results are strongest: In this situation, the
binomial ideals are parametrized by characters on sublattices of the free
abelian group whose group algebra is the center of $\mathcal{T}_{\bf{q}}$; the
sublattice-character pairs corresponding to primitive ideals as well as to
radicals and minimal primes of binomial ideals are determined. As for
occurrences of binomial ideals in quantum algebras: It is shown that
cocycle-twisted group algebras of finitely generated abelian groups are
quotients of quantum tori modulo binomial ideals. Another appearance is as
follows: Cocycle-twisted semigroup algebras of finitely generated commutative
monoids, as well as quantum affine toric varieties, are quotients of quantum
affine spaces modulo certain types of binomial ideals.
