Question

Show that the center Z(R) of prime ring R is an integral domain, and char R is either 0 or a prime number.

Answer 1

Let nonzero a ∈ Z(R), and say ab = 0. Then aRb =Rab = 0, so b = 0 since R is a prime ring. This says that ais not a zero-divisor in R, and for domains char R is either0 or a prime number.