I was reading a paper, and it said that the following were equivalent using the Axiom of Choice, but I tried working it out, and I wasn't sure how: an algebra $A$ is primitive; $A$ has a proper left ideal $B$ such that $A = B +C$ for any non-trivial two-sided ideal $C$ of $A$. I've tried reasoning it out, and I'm not sure how - can anyone help? I know it should be really easy, but I seem to be missing a key step. If someone could give me a hint or two to work it out, that would be great.
Lam (A first course in noncommutative rings, 2ed) does it for (unital) rings $R$ in Lemma 11.28 (page 186):
Here, the statement equivalent to the Axiom of Choice is Zorn's Lemma; it says that:
In this proof we get to see one of its most common uses: it assures that any unital ring has a maximal ideal (see the Wikipedia page for more information).