  Pursuing ideas of Jeff Smith, we develop a homotopy theory of ideals of
monoids in a symmetric monoidal model category. This includes Smith ideals of
structured ring spectra and of differential graded algebras. Such Smith ideals
are NOT subobjects, and as a result the theory seems to require us to consider
all Smith ideals of all monoids simultaneously, rather then restricting to the
Smith ideals of one particular monoid. However, we can take a quotient by a
Smith ideal and get a monoid homomorphism. In the stable case, we show that
this construction is part of a Quillen equivalence between a model category of
Smith ideals and a model category of monoid homomorphisms.
