[seqfan] Is A037289 multiplicative

[franktaw at netscape.net]

Looking at this sequence, there is a conjecture that the sequence is 
multiplicative. I can almost prove that it is.

I need a lemma that, if R is a commutative ring of order n, and prime p 
divides n, then there is a Sylow p-subgroup of the additive group of R 
that is in fact a subring if R. It is then easy to see that R must be 
the direct product of these subrings, and the result follows.

(This is an equivalent assertion, by the way; if there is a commutative 
ring with no such "Sylow p-subring", the sequence is not multiplicative 
for that n.)

Does anybody know of such a result? Or can you find a proof?

Franklin T. Adams-Watters