  We describe the formal properties of cellularization functors in triangulated
categories and study the preservation of ring and module structures under these
functors in stable homotopy categories in the sense of Hovey, Palmieri and
Strickland, such as the homotopy category of spectra or the derived category of
a commutative ring. We prove that cellularization functors preserve modules
over connective rings but they do not preserve rings in general (even if the
ring is connective or the cellularization functor is triangulated). As an
application of these results, we describe the cellularizations of Eilenberg-Mac
Lane spectra and compute all acyclizations in the sense of Bousfield of the
integral Eilenberg-Mac Lane spectrum.
