## A ring has only the trivial one-sided ideals precisely when it is a division ring

Let $R$ be a ring with $1 \neq 0$. Prove that $R$ is a division ring if and only if the only left ideals of $R$ are $0$ and $R$. (The statement remains true when “left” is replaced by “right”.)

$(\Rightarrow)$ Suppose $R$ is a division ring. Let $L \subseteq R$ be a nonzero left ideal; that is, $rL \subseteq L$ for all $r \in R$. Now let $a \in L$ be nonzero. Since $R$ is a division ring, there exists $b \in R$ such that $ba = 1 \in L$. Then for all $r \in R$, $r \cdot 1 \in L$, and in fact $L = R$. Thus the only left ideals of $R$ are $0$ and $R$.

$(\Leftarrow)$ Suppose the only left ideals of $R$ are $0$ and $R$, and let $a \in R$ be nonzero. Now the left ideal $Ra$ generated by $a$ is nonzero since it contains $a$, so that $Ra = R$. So there exists an element $b \in R$ such that $ba = 1$. Similarly, there exists $c \in R$ such that $cb = 1$. Now $bab = b$, so that $cbab = cb$, and hence $ab = 1$. So $a$ is a unit in $R$, and thus $R$ is a division ring.