Two sided ideals are maximal right ideal iff they are maximal left ideal.

Question

Let $R$ be a ring with unity and $I$ be a two-sided ideal in $R$. Then $I$ is a maximal right ideal if and only if it is a maximal left ideal.

Would anyone give me an idea to prove the statement? Thanks.

Answer 1

Suppose $R/I$ only has trivial right ideals. Then it has only trivial left ideals. For if $x$ is a nonzero member of $R/I$, $x(R/I)=R/I$, and $x$ is right invertible, say by element $y$ similarly $y$ is right invertible, say by element $z$, but it is any easy exercise to prove $x=z$, so $x$ is a unit and $R/I$ is a division ring, and therefore only has trivial left and right ideals.

By a symmetric argument, the words left and right can be interchanged.