Jacobson Radical $J(R)$ is a proper ideal I found a remark on my notes:

Jacobson Radical $J(R)$ is a proper ideal. Hint: Zorn's Lemma

I know $J(R)$ is the intersection of all the maximal left ideals of ring $R$. I know the maximal ideals are proper by definition. However, in the remark i guess it must be a two sided ideal. 
But, we have showed that $J(R)$ is two sided by the claim that it is the intersection of all annihilators of all simple $R$-modules which are two sided ideals.

Back to my remark, how come Zorn's Lemma is on the table? Could you please enlighten me?

This also may help .
If M is an R-module, then Jacobson radical J(M)= $J_{R}$(M) of R-module M is the intersection of all maximal submodules of M. (Maximal submodules mean maximal proper submodules ).
So if M is finitely generated, then every submodule $N$ of M is contained in a maximal submodules, by Zorn's Lemma. (Otherwise,  If the union of a chain of proper submodules is equal to M, then the union contains all the generators, hence some member of the chain contains all the generators, contradiction.) Now, take $N=0$, so we get J(M) is a proper submodule of M. Since R is finitely generated,  J(R) is always proper left ideal.  
