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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