I am reading about torsion of a module over a commutative ring.
Definition: $\text{Tor(M)} = \{ x \in M | rx = 0 \text{for some non zero r} \in R $ where $R$ is an integral domain. This is a submodule.
I was wondering what can be said in the case when $R$ is a non commutative domain? In general in the case of a non commutative domain we cannot say that $\text{Tor(M)}$ is a submodule because if $x_1, x_2 \in \text{Tor(M)}$ then it is not clear what annihilates $x_1+x_2$.
However I read on Wikipedia that when $R$ is an Ore ring then $\text{Tor(M)}$ is a submodule.
A non-commutative domain is called an Ore ring if for any two non-zero elements $r_1$ and $r_2$ there exists $r_3, r_4$ such that $r_1r_3 = r_2r_4$. This condition is very naturally arrived at in triyng to give a mulitplicative structure to the field of fractions of a non-commutative domain.
But I am unable to see how to use it prove that $\text{Tor(M)}$ is a submodule.
Please help. Thanks!
Best Answer
Hint: If $x_1r_1=0=x_2r_2$, then $r_1r_3=r_2r_4\in\operatorname{Ann}_R(x_1+x_2)$.