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

Question

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

Answer 1

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