# Facts about Ideals in Non-Commutative Rings

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:

1. Let $$R$$ have unique two-sided maximal ideal $$I$$. Must $$I=J$$?
2. 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 thoughts:

1. My lecturer seemed to assert this, but I don't see why this should be the case.
2. 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 $$J' \not\subset J$$?
3. 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?

Definitions: https://en.wikipedia.org/wiki/Maximal_ideal

1. No. Take $$V$$ to be a countable dimensional $$F$$-vector space, and consider the ring $$End(V_F)$$.
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?