division ring means no non trivial left ideals

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.

Re: division ring means no non trivial left ideals

Quote:

Originally Posted by

**abhishekkgp** 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.

Re: division ring means no non trivial left ideals

Re: division ring means no non trivial left ideals

tl,dr; version: Ra is a left ideal for all a, so 1 = sa, for some s in R

(there is a few gaps in this, of course)

Re: division ring means no non trivial left ideals

Quote:

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

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.

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.

Quote:

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??

your argument cannot possibly be right. is a ring with no zero divisors, and it definitely DOES have non-trivial ideals.

Re: division ring means no non trivial left ideals

Quote:

Originally Posted by

**Deveno** 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.

By i meant .

I got my mistake. My argument works only if is finite. Do you agree?

Re: division ring means no non trivial left ideals

yes, because then we know that aR ≠ R means that the map r --> ar is not injective. with infinite rings, we can make no such claim.