I am following Lams book "A first course in non-commutative rings". I am attempting to prove that the Jacobson radical of a ring is precisely the intersection of all maximal modular left ideals of said ring (following the exercises on page 63 -64).
I am almost done, the only thing I need to do is prove that the Jacobson radical is contained in every maximal modular left ideal. On this front I am stuck.
How does one prove this?