If R is a unique factorization domain and if a,b are in R, then a and b have a least common multiple (l.c.m.) in R.

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• If R is a unique factorization domain and if a and b in R are relatively prime (i.e.,(a,b) = 1), whenever a divides bc, then a divides c. That is, if R is a unique factorization domain and if a and b in R are relatively prime (i.e., (a,b) = 1), whenever a divides bc then a divides c.

  132134 If R is a unique factorization domain and if a and b in R are relatively prime (i.e.,(a,b) = 1), whenever a divides bc, then a divides c.

• Unique Factorization Domain with Quotient Field

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  Since R is an unique factorization domain (UFD), then we have unique factorizations for both a and b a = p_1 * p_2 * ... * p_k b = q_1 * q_2 * ... * q_t where p_1, ..., p_k are prime factors of a, q_1, ..., q_t are prime factors of b Then ab = p1

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  Since R is a principal ideal domain, then we can find some d in R, such that I = = <a,b>. Then a is in I = and this implies that d|a; b is in I = and this implies d|b. So d is a common divisor of a and b.