Question #17276

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.

Expert's answer

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.

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