In a PID, two ideals are comaximal precisely when 1 is a gcd of their generators

Prove that in a principal ideal domain, two ideals (a) and (b) are comaximal (that is, (a) + (b) = R if and only if 1 is a greatest common divisor of a and b. In this case, we say that a and b are relatively prime.


Suppose (a) + (b) = R. Since R has a 1, we have (a) + (b) = (a,b) = (1). Then by Proposition 6 in D&F (page 280), 1 is a greatest common divisor of a and b.

Conversely, suppose 1 is a greatest common divisor of a and b. Then we have ax + by = 1 for some x,y \in R, so that 1 \in (a) + (b), and hence (a) + (b) = R.

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