Suppose that is a ring with no proper left ideals. If has then is a division ring. To see this, let Then and so for some Since we have and hence for some Then and so proving that is a division ring.

But what if doesn’t have The following problem answers this question.

**Problem**. Let be a ring, which may or may not have Show that if has no proper left ideals, then either is a division ring or and for some prime number

**Solution**. Let

Then is a left ideal of because it’s clearly a subgroup of and, for and we have and so i.e. So either or

Case 1: That means for all or, equivalently, Thus every subgroup of is a left (in fact, two-sided) ideal of Hence has no proper subgroup (because has no proper left ideals) and thus for some prime

Case 2: Choose So and hence because is clearly a left ideal of Thus there exists such that Now

the left-annihilator of in , is obviously a left ideal of and we can’t have because then So Since

we have Thus Let

Clearly is a left ideal of and Thus So for all Now let be any element of Then, by what we just proved, On the other hand, by the same argument we used for we find such that Thus i.e.

So and hence i.e. and thus

So and hence for all Thus proving that is a division ring.

**Remark**. The same result given in the above problem holds if has no proper right ideals.

**Example**. Let be a prime number. The ring

is not a division ring and it has no proper left (or right) ideals.