Let $R$ be a unital, but not necessarily commutative ring. Define its Jacobson radical $J$ to be the intersection of all maximal left ideals of $R$. Three questions:
- Let $R$ have unique two-sided maximal ideal $I$. Must $I=J$?
- Define $J'$ to be the intersection of all maximal two-sided ideals of $R$. Is $J' = J$?
3(a). Is every maximal left ideal a maximal ideal?
3(b). Is every maximal ideal a maximal left ideal?
- My lecturer seemed to assert this, but I don't see why this should be the case.
- My guess is no. It does not even seem that we have an inclusion either way. Are there matrix rings that show $J \not\subset J'$ and $J' \not\subset J$?
- I think no to both. (If the answers were yes, then item 2 would become trivial.)
3(a). No. A counterexample is: let $R$ be a matrix ring over a division ring $D$. Then $R$ is a simple ring; however it has a nonzero maximal left ideal.
3(b). I guess no, because even though a maximal two-sided ideal $K$ is a left ideal, and maximal among two-sided ideals, I don't see a reason why it should be maximal among left ideals. UPDATE: Actually, I'd like to correct my guess to 'yes'. Using Zorn's Lemma, a maximal left ideal $K'$ containing $K$ is contained in a maximal ideal $L$. Then $K=L$ (since $K$ is a maximal two-sided ideal and $L$ is proper two-sided-sided), which forces $K=K'$. Is this argument correct?