TITLE: Multiplicative Structures on Moore Spectra
QUESTION [20 upvotes]: The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the Moore spectrum. For concreteness this is defined as a spectrum $X$ such that $\pi_n(X) = 0$ for $n <0$ $H_n(X)= 0$ for $n >0$ and $H_0(X) = R$ for some ring $R$. 
There are some curious phenomenon that happens:

On one extreme, the Mod 2 Moore spectrum has no unital multiplication at all (by simple arguments in, say, Difficulties with the mod 2 Moore Spectrum)
The Mod 3 Moore spectrum is not $A_{\infty}$ by Massey product arguments.
The comment here on top of page 838: http://www.math.uni-bonn.de/people/schwede/rigid.pdf says that the mod $p$ Moore spectrum for $p \geq 5$ is homotopy associative by folklore (I would like to see an argument for this too!)
On another extreme, since we can model the $\mathbb{Z}[q^{-1}]$ by localizing the sphere spectrum they are $E_{\infty}$.

In this light, my questions are:

First and foremost, I would love to see a proof of the folklore result above about $p \geq 5$
Is there a "general pattern" about multiplicative structures of the Moore spectrum as the ring/abelian group varies

REPLY [5 votes]: The classical reference is as follows:
@article {MR760188,
    AUTHOR = {Oka, Shichir{\^o}},
     TITLE = {Multiplications on the {M}oore spectrum},
   JOURNAL = {Mem. Fac. Sci. Kyushu Univ. Ser. A},
  FJOURNAL = {Memoirs of the Faculty of Science. Kyushu University. Series
              A. Mathematics},
    VOLUME = {38},
      YEAR = {1984},
    NUMBER = {2},
     PAGES = {257--276},
      ISSN = {0373-6385},
     CODEN = {MFKAAF},
   MRCLASS = {55P45},
  MRNUMBER = {760188 (85j:55019)},
MRREVIEWER = {Donald M. Davis},
       DOI = {10.2206/kyushumfs.38.257},
       URL = {http://dx.doi.org/10.2206/kyushumfs.38.257},
}

Here is the review by Don Davis:

For any positive integer $q$, let $M_q$ denote the Moore spectrum
  whose only nontrivial homology group is $\mathbb{Z}/q$ in dimension 0.
  The main result is the following theorem: (a) The number of homotopy
  classes of multiplications on $M_q$ is $4$ if $q≡0 (4)$, 1 if $q$ is
  odd, and $0$ if $q≡2 (4)$. (b) These multiplications are commutative if
  and only if $q≡0$ (8) or $q$ is odd, and are associative if and only
  if $q\not\equiv 2 (4)$ and $q\not\equiv\pm 3 (9)$.     A number of
  more technical results are proved, many involving premultiplications
  and regularity.