  We investigate the structure of ideals generated by binomials (polynomials
with at most two terms) and the schemes and varieties associated to them. The
class of binomial ideals contains many classical examples from algebraic
geometry, and it has numerous applications within and beyond pure mathematics.
The ideals defining toric varieties are precisely the binomial prime ideals.
Our main results concern primary decomposition: If $I$ is a binomial ideal then
the radical, associated primes, and isolated primary components of $I$ are
again binomial, and $I$ admits primary decompositions in terms of binomial
primary ideals. A geometric characterization is given for the affine algebraic
sets that can be defined by binomials. Our structural results yield
sparsity-preserving algorithms for finding the radical and primary
decomposition of a binomial ideal.
