2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221916M. Mahowald, in his work on $b{\rm o}$-resolutions, constructed a $b{\rm o}$-module splitting of the spectrum $b{\rm o} \wedge b{\rm o}$ into a wedge of summands related to integral Brown-Gitler spectra. In this paper, a similar splitting of $b{\rm o} \sm tmf$ is constructed. This splitting is then used to understand the $b{\rm o}_*$-algebra structure of $b{\rm o}_* tmf$ and allows for a description of $b{\rm o}^* tmf$.15 pages, 4 figuresAlgebraic Topology55P10 (Primary) 55P42, 55Q51 (Secondary)text