How do I prove these two definitions for a local ring are equivalent?

Question

Let $R$ be a non-trivial ring with unity. We say $R$ is a local ring if there exists a unique maximal left-ideal.

Let $R$ be a local ring. Then, how do I prove that there exists a unique maximal right-ideal?

I know that the maximal left-ideal is the Jacobson radical and it is the intersection of all the maximal right-ideals. However, I am not sure if this helps to prove the problem.

Answer 1

There are at least two 'symmetric' characterizations of local rings available:

• For all $x\in R$, at least one of $x$ and $1-x$ is a unit
• The set of nonunits is closed under addition.

You should be able to establish the equivalence of at least one of these with your definition, and then it is automatic.

Answer 2

One way to start your proof is to suppose $K$ is some maximal right ideal. You know that $K$ contains $J$, since $J$ is the intersection of all the right ideals. (Incidentally, I'm assuming you know that there is at least one right ideal, since $J$ is the intersection of them and $J$ is non-degenerate). Now, consider any element $k \in K$. Then show that $Nk+J:=\{nk+j:n\in N, j\in J\}$ is a subgroup of the additive group of $R$, where $nk=k+k+...+k$ if $n>=0$, and $(-n)k=n(-k)$. Then show that $Nk+J$ is a left ideal of $R$. Lastly, draw some conclusions about $k$ and $K$ in relation to $J$.