Is it true that the Moore spectrum for the group $\mathbb{Z}_{(p)}$ can be constructed by smashing $\mathbb{S}$ with $q^{-1}\mathbb{S}$ for each $q\neq p$ (here both $q$ and $p$ are primes). It seems we might wish to show this by showing that $[\mathbb{S},\mathbb{S}_{(p)}\wedge H\mathbb{Z}]_\ast\cong[\mathbb{S},H\mathbb{Z}_{(p)}]$ but I cannot see how to show that either.

Thanks for any help on this matter!