Question

For any semiprime ring R, show that Z(R) is reduced, and that char R is either 0 or a square-free integer.

Answer 1

Let a ∈ Z(R) be such that a^2 = 0. Then aRa= Ra^2 = 0, so a = 0. This shows that Z(R) is areduced (commutative) ring, and then char R is either 0 or a square-freeinteger.