  We construct Quillen equivalences between the model categories of monoids
(rings), modules and algebras over two Quillen equivalent model categories
under certain conditions. This is a continuation of our earlier work where we
established model categories of monoids, modules and algebras [Algebras and
modules in monoidal model categories, Proc. London Math. Soc. 80 (2000),
491-511]. As an application we extend the Dold-Kan equivalence to show that the
model categories of simplicial rings, modules and algebras are Quillen
equivalent to the associated model categories of connected differential graded
rings, modules and algebras. We also show that our classification results from
[Stable model categories are categories of modules, Topology, 42 (2003)
103-153] concerning stable model categories translate to any one of the known
symmetric monoidal model categories of spectra.
