Maximal right ideals

Question

A question on basic understanding.

If $R$ is a ring and $I$ is a maximal two-sided ideal of $R$, then we know $R/I$ is a field. My question is what, if anything, can we conclude if $I$ is merely a maximal one-sided ideal? Do we obtain a division ring?

Answer 1

If $R$ is a ring and $I$ is a maximal two-sided ideal of $R$, then we know $R/I$ is a field.

No, that is not true. You only know it is a simple ring, which means it has exactly two two-sided ideals (both trivial.)

what, if anything, can we conclude if $I$ is merely a maximal one-sided ideal?

If $I$ is a maximal right ideal, you can say that $R/I$ is a simple right $R$ module, which means it has exactly two submodules (both trivial.)

If you have a two-sided $I$ ideal which happens to be maximal as a right ideal, then you can say that $R/I$ is a division ring.