The result should follow more easily if you know that if a monoid has left inverses for all its elements its actually a group. More information can be found on my blog, here.
Prove that is a division ring if and only if the only left ideals in are and .(I am NOT assuming that is commutative)
I can prove that if is a division ring then every left ideal is trivial.
I cannot prove it in the other direction. Help needed on this.
The result should follow more easily if you know that if a monoid has left inverses for all its elements its actually a group. More information can be found on my blog, here.
I messed up. was using wrong definition of division ring.
Anyways, Can you please check whether the following is true:
If is a ring with with no zero divisors then each non-zero element of has a left inverse and a right inverse.
Proof: We claim that .
If this is not the case then such that which contradicts that has no zero divisors.
So .
Therefore there exists with .
Similarly for left inverse.
Does the above also not prove that if is a ring with with no zero divisors then every left ideal of is trivial??
why are you using aR? this is not a left ideal, generally. i can't see how you can say that and exist with , either.
your argument cannot possibly be right. is a ring with no zero divisors, and it definitely DOES have non-trivial ideals.So .
Therefore there exists with .
Similarly for left inverse.
Does the above also not prove that if is a ring with with no zero divisors then every left ideal of is trivial??