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.
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.