Is every simple ring a division ring?

Question

I know that every division ring is simple.

Is the converse true?

I think it isn't. But I can't find a counterexample.

Answer 1

Well, most of the time when two concepts are given names, as "division ring" and "simple ring" are, the ideas are different. If every simple ring were a division ring, then we would have created an extra unnecessary piece of terminology.

You could be forgiven for not finding a counterexample if you were only thinking of commutative rings though. It does turn out that a commutative simple ring is a field.

Anyhow, there are many good noncommutative rings which are simple. The most obvious one are, as has been mentioned, full $n\times n$ matrix rings over division rings (with $n>1$ .)

Another collection of examples is given by Weyl algebras, which in general aren't division rings but are nevertheless simple rings and additionally they don't have any zero divisiors other than $0$.

You can produce simple rings from noncommutative rings by forming their quotient by a maximal ideal of the ring. For example, if we take $R$ to be the ring of linear transformations of a vector space with countably infinite dimension, it is known that ring has exactly one proper ideal $J$ (which is obviously then maximal.) Then then ring $R/J$ is simple. It turns out that it has zero divisors but isn't Noetherian or Artinian, so this example is different from the first two I gave.

Answer 2

Note: I did not read the question carefully enough, as Prahlad Vaidyanathan points out in the comments.

Let $R$ be a division ring and let $L \subset R$ be a nonzero left ideal. Let $x \in L$ be nonzero. Since $R$ is a division ring, $x$ has a 2-sided inverse $x^{-1} \in R$. But then, $1 = x^{-1} \cdot x \in L$. Therefore, $y = 1 \cdot y \in L$ for every $y \in R$ and so $L = R$. So, $R$ does not even have a nontrivial (i.e. nonzero and proper) left ideal.

The whole point is that, as soon as an ideal contains a unit of a ring, it eats up the whole ring. Since division rings contains nothing but units (and zero), a nontrivial (nonzero and proper) ideal is impossible.