The Hopf ring for bo and its connective covers

In this work we recompute the known ordinary image homology of the spectrum bo as a Hopf ring. In addition, our study computes the Hopf modules over the Hopf ring image for the connective covers boleft angle bracket1right-pointing angle bracket,boleft angle bracket2right-pointing angle bracket, and boleft angle bracket4right-pointing angle bracket. These results are calculated using the bar spectral sequence, which allows us to advance from one space to the next in the spectrum, and by using maps from our spectra to the image Eilenberg–Mac Lane spectrum image in order to facilitate the simplification of our result. These calculations yield, in turn, the Hopf ring for boleft angle bracketnright-pointing angle bracket for all n.