  We investigate the relationship between differential graded algebras (dgas)
and topological ring spectra. Every dga C gives rise to an Eilenberg-Mac Lane
ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and
D are topologically equivalent. Quasi-isomorphic dgas are topologically
equivalent, but we produce explicit counter-examples of the converse. We also
develop an associated notion of topological Morita equivalence using a
homotopical version of tilting.
