Before revision:  Injective chemical reaction networks do not have the capacity to admit multiple positive steady states for any rate constants and within each stoichiometric class.
After revision:  Injective chemical reaction networks do not have the capacity to admit multiple positive steady states within any stoichiometric class for any choice of rate constants. Our work extends previous results by Craciun and Feinberg for chemical reaction networks with stoichiometric space of maximal dimension. Specifically, a network is called injective if the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class.