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

Question #23254

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

Expert's answer

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.
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