Answer to Question #23254 in Abstract Algebra for Hym@n B@ss

Question

Show that a ring R is a domain iff R is prime and reduced

Answer 1

First assume R is a domain.Then a^n = 0 =⇒ a = 0, so R is reduced. Also, aRb= 0 ⇒ ab = 0 ⇒ a = 0 or b= 0, so R is prime. Conversely, assume R is prime andreduced. Let a, b ∈ R be suchthat ab = 0. Then, for any r ∈ R, (bra)^2 = br(ab)ra = 0, so bra =0. This means that bRa = 0, so b = 0 or a = 0, since R isprime.