# Answer to Question #17105 in Algebra for john.george.milnor

Question #17105

Show that the center of a simple ring is a field, and the center of a semisimple ring is a finite direct product of fields

Expert's answer

Suppose

where the

*R*is a simple ring,and let*a**∈**Z*(*R*).Then*Ra*is an ideal, so*Ra*=*R*. This implies that*a**∈**U(**R*). But clearly*a−*1*∈**Z*(*R*), so*Z*(*R*) is a field. Next, assume*R*is asemisimple ring, and let*R*= (direct product on i=1 to i=r)M*ni*(*Di*)where the

*Di*’s are divisionrings. Z(*D*) = (direct product on i)Z*(M**ni*(*Di*))*∼*(direct product on i)Z(*Di*)*,*where the*Z*(*Di*)’s are fields.
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