Equivalent definitions of the Jacobson Radical Noncommutative Algebra / Farb & Dennis defines the Jacobson Radical $J(R)$ of a ring $R$ as the intersection of all annihilators of simple $R$-modules (page 58).
It is then claimed that $J(R)$ is also the intersection of all maximal left ideals in $R$.
The given explanation is that "simple $R$-modules are precisely $R/I$ for maximal left ideals $I$".
I've been thinking about it and I'm confused. I do believe that a simple $R$-module is isomorphic to $R/I$ for some maximal ideal $I$, but it doesn't mean that it actually is $R/I$.
Is there really a problem here? If so, how can it be fixed?
 A: Corrected with thanks to Cihan for the pertinent comments to this question.
It should have said "the annihilators of simple $R$ modules are precisely the annihilators of modules $R/I$ for maximal left ideals $I$, and these annihilators are intersections of the annihilators of their individual nonzero elements; the latter annihilators are all maximal left ideals, among which $I$ itself as the annihilator of the element $1+I\in R/I$". The first part is because every simple module is isomorphic to some $R/I$, and isomorphic modules have identical annihilators.
I would also like to note that the definition "the intersection of all annihilators of simple $R$-modules" is unnecessarily referring to the class of all simple $R$-modules, which is too large to be called a set. Strictly speaking this is forming a class of sets, whose intersection is taken. One can indeed define the intersection of any non-empty class of sets, but in this context this seems set-theoretic overkill to me: for one thing the annihilators being intersected actually form a set, because of the above equivalence and also because they are all subsets of $R$ (the set of annihilators is a subset of the powerset of $R$). Saying "the intersection of all maximal left ideals in $R$" would be a much clearer as definition to me (and is heavy enough definitional artillery as it stands).
However as Cihan noted, this description does not make it obvious that the radical is a two-sided ideal, whereas the annihilator description does make this obvious (to those who, unlike I did when I first wrote this answer, know enough about such matters).
