TITLE: Is a p-Sylow subgroup of the underlying additive group of a finite commutative ring an ideal?
QUESTION [9 upvotes]: Let $R$ be a finite commutative ring. Let $H$ be a $p$-Sylow subgroup of the additive group $(R,+)$. Is $H$ an ideal in $R$?

For ex: 
R= Z/6Z is a finite commutative ring. 
H={0,3} is a 2-sylow subgroup which in itself is an ideal of the ring.
Is this true for any finite commutative ring R?

REPLY [3 votes]: Let $H$ be a $p$-Sylow subgroup of $(R,+)$ ; let$|R|=p^mn$ , where $p$ does not divide $n$  ; then 
$|H|=p^m$. As $(R,+)$ is abelian , so $H$ is normal in $(R,+)$ and $|R/H|=n$ . Now let $r\in R , a\in H$ ,
then $ra+H \in (R/H,+)$ , so $nra+H=n(ra+H)=H$ , so $nra \in H$ . Now as 
g.c.d.$(p^m,n)=1 $ , so $\exists s,t \in \mathbb Z$ such that $sp^m+tn=1$ , so 
$ra=t(nra)+s(p^mra)=t(nra)+sr(p^ma)=t(nra)\in H$ , where $p^ma=0$ because $a\in H $ ,
and $ |H|=p^m$ . Thus $H$ is an ideal