Binomials are polynomials with at most two terms. A binomial ideal is an ideal generated by binomials. Primary components and associated primes of a binomial ideal are still binomial over algebraically closed fields. Primary components of general binomial ideals over algebraically closed fields with characteristic zero can be described combinatorially by translating the operations on binomial ideals to operations on exponent vectors. In this dissertation, we obtain more explicit descriptions for primary components of special binomial ideals. A feature of this work is that our results are independent of the characteristic of the field.
First of all, we analyze the primary decomposition of a special class of binomial ideals, lattice ideals, in which every variable is a nonzerodivisor modulo the ideal. Then we provide a description for primary decomposition of lattice ideals in fields with positive characteristic.
In addition, we study the codimension two lattice basis ideals and we compute their primary components explicitly.
An ideal I ? k[x_(1),....x_(n) ] is cellular if every variable is either a nonzerodivisor modulo I or is nilpotent modulo I. We characterize the minimal primary components of cellular binomial ideals explicitly. Another significant result is a computation of the Hull of a cellular binomial ideal, that is the intersection of all of its minimal primary components.
Lastly, we focus on commutative monoids and their congruences. We study properties of monoids that have counterparts in the study of binomial ideals. We provide a characterization of primary ideals in positive characteristic, in terms of the congruences they induce.
