Question

Let R be a J-semisimple domain and a be a nonzero central element of R.
Show that the intersection of all maximal left ideals not containing a is zero.

Answer 1

Let x be an element in thisintersection. We claim that ax ∈ rad R. Once we have provedthis, the hypotheses on R imply that ax = 0 and hence x =0. To prove the claim, let us show that, for any maximal left ideal m, we have ax∈ m. If a ∈ m, this is clear since a ∈ Z(R). If a is not in m, then bythe choice of x we have x ∈ m, andhence ax ∈ m.