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November 14th, 2013, 09:56 AM   #1
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Division Ring

Let R be a ring with 1 . Prove that R is a division ring if and only if the only left ideals of R are 0 and R.
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November 14th, 2013, 02:47 PM   #2
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Re: Division Ring

Quote:
Originally Posted by sebaflores
Let R be a ring with 1 . Prove that R is a division ring if and only if the only left ideals of R are 0 and R.
Suppose 1 is an element of an ideal in a division ring. What can you say about the ideal? That's a hint to get you thinking in the right direction.

(edit) No response for a while, here's more of a hint.

A left ideal is characterized by the property that it's closed under arbitrary (left) multiplication by any element of the ring R. I think of ideals as "absorbing" multiplication.

So suppose a left ideal X contains 1. For any element r of the ring R, r1 = r is in X. So X contains everything in R, in other words X = R. So if X contains 1 it's the entire ring.

Ok that's the background. To refresh our memories and provide clarity, here is the original problem.

Quote:
Originally Posted by sebaflores
Let R be a ring with 1 . Prove that R is a division ring if and only if the only left ideals of R are 0 and R.
Ok we have an if and only if, so there are two directions. We have to prove both the => and the <= directions of the bi-implication.

(=>) Let R be a division ring and show that its only left ideals are R and 0.

Ok, suppose we have an ideal X. If it consists of only 0 we're done. So there's some x in X that's not zero.

Since R is a division ring, if x is not zero then 1/x exists in R. But we can multiply the elements of X by any elements of R, including 1/x.

But now 1/x * x = 1, so 1 is in X. And what do we know happens the moment we know that 1 is in an ideal? We know that the ideal is the whole ring. Because if 1 is in X then so is r1 for any r in R.

So that's the (=>) direction.

Next, the other direction.

(<=)

Suppose that R is a ring such that its only ideals are R and 0. We must prove that R is a division ring.

Do you want to take a shot at carrying this forward?
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