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 *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.

where the

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