Maximal ideals in rings

Question

Let $R$ be a ring with identity. Is it true that if $R$ has a finite maximal right ideal then it MUST have a finite maximal two-sided ideal ?

Answer 1

This is actually a question about monoids, since the additive structure does not help.

Let $I$ be a finite maximal right ideal of a monoid $M$ and let $x \in MI$. Then $x = uv$ for some $u \in M$ and $v \in I$. Let $J = (\{x\} \cup I)M$. Then $J$ is a right ideal containing $I$. Further, since $xM = uvM \subseteq uI$, we get $J \subseteq uI \cup I$ and hence $J$ is finite. By maximality, $J = I$ and in particular, $x \in I$. Thus $MI = I$ and $I$ is a finite two-sided ideal. It is of course a finite maximal two-sided ideal, since it is already maximal among the larger class of finite right ideals.