0

Let $R$ a unital ring (not necessarily commutative) and $M$ a $R-$module. Let $$Tor(M)=\{m\in M\mid \exists r\in R, r\neq 0: rm=0\}.$$

1) Show that if $R$ is a domain then $Tor(M)$ is a $R-$submodule.

2) Show that if $R$ is not a domain, $Tor(M)$ is not a module.

Attempts:

For 1) let $m,n\in Tor(M)$. Then there is $r,s\in R$ s.t. $rm=0$ and $sn=0$. I wanted to have $rs(m+n)=0$, but since $R$ is not commutative, I can't conclude.

For 2) I have no idea.

user386627
  • 1,828

1 Answers1

1

In 1) $R$ is a domain, thus in particular commutative. Note that is is false if $R$ is not commutative, see here.

For 2) consider $R=M=\mathbb Z/6\mathbb Z$. $2$ and $3$ are torsion, but $5$ is not.

MooS
  • 31,390
  • Every domain is commutative ? In my definition, a ring $R$ is a domain if $R$ has no divisor of $0$. – user386627 May 17 '17 at 08:29
  • I do not know your definition, but 1) is false without the commutative assumption, so it is probably meant to be commutative. – MooS May 17 '17 at 08:38
  • Here is an example, that it is false without the commutative assumption: https://math.stackexchange.com/questions/1939173/torsion-elements-do-not-form-a-submodule – MooS May 17 '17 at 08:43