If a non-trivial ring $R$ has a unique maximal left ideal $J$ , then $J$ is
two-sided and is also the unique maximal right ideal in $R$.
I can prove that it is two sided, but I can't prove that it is unique.
My proof:
Let $r \in R$. Then $Jr$ is a left ideal. If $Jr = R$, then $jr = 1$ for some $j \in J$. Note that $rj \in J$ since J is a left ideal, so $rj \neq 1$. Using a lemma, $1 – rj$ is not left invertible, which is to say that $R(1-rj) \neq R$. But then R(1-rj) is contained in a maximal left ideal, i.e. $1-rj \in R(1-rj) \subseteq J$, so $1= rj +(1-rj) \in J$, which gives a contradiction. Hence $Jr \neq R$. This implies that $Jr$ is contained in $J$. Since the choice of $r$ is arbitrary, $jr \in J$ for all $j \in J$ and $r \in R$. Hence, $J$ is a two sided ideal.
Can someone tell me how I can prove uniqueness?
Best Answer
Suppose there's another right ideal $J'$ such that $J' \not \subseteq J$. Take $x \in J'$ such that $x \notin J$. Then $x$ is left invertible, so there exists $r \in R$ such that $rx =1$. But $xr \neq 1$. Thus both $xr $ and $1 -xr$ are not left invertible (by your lemma), thus $xr, 1 - xr \in J$, so $1 \in J$, contradiction. Therefore $J$ is the unique maximal right ideal.