# Answer to Question #17276 in Algebra for sanches

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.

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.*Need a fast expert's response?

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