Originally Posted by

**abhishekkgp** I messed up. was using wrong definition of division ring.

Anyways, Can you please check whether the following is true:

If $\displaystyle R$ is a ring with $\displaystyle 1 \neq 0$ with no zero divisors then each non-zero element of $\displaystyle R$ has a left inverse and a right inverse.

Proof: We claim that $\displaystyle aR=R,a \in R, a \neq 0$.

If this is not the case then $\displaystyle \exists r_1,r_2 \in R, r_1 \neq r_2, r_1 \neq 0, r_2 \neq 0$ such that $\displaystyle ar_1=ar_2 \Rightarrow a(r_1-r_2)=0$ which contradicts that $\displaystyle R$ has no zero divisors.