<p>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.
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Thanks for any help on this matter. I apologize if this question is too basic.  I have asked it on MSE and not recieved any help.
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