  Chemical reaction networks are often used to model and understand biological
processes such as cell signaling. Under the framework of chemical reaction
network theory, a process is modeled with a directed graph and a choice of
kinetics, which together give rise to a dynamical system. Under the assumption
of mass action kinetics, the dynamical system is polynomial. In this paper, we
consider the ideals generated by the these polynomials, which are called
steady-state ideals. Steady-state ideals appear in multiple contexts within the
chemical reaction network literature, however they have yet to be
systematically studied. To begin such a study, we ask and partially answer the
following question: when do two reaction networks give rise to the same
steady-state ideal? In particular, our main results describe three operations
on the reaction graph that preserve the steady-state ideal. Furthermore, since
the motivation for this work is the classification of steady-state ideals,
monomials play a primary role. To this end, combinatorial conditions are given
to identify monomials in a steady-state ideal, and we give a sufficient
condition for a steady-state ideal to be monomial.
