In an integral domain, two principal ideals are equal precisely when their generators are associates

Let R be an integral domain and let a,b \in R. Prove that (a) = (b) if and only if a = ub for some unit u \in R.

(\Rightarrow) Suppose (a) = (b). Then a \in (b), and we have a = ub for some element u \in R. Similarly, b \in (a) and we have b = va for some element v \in R. Now a = uva, and since R is a domain, we have uv = vu = 1. Thus u is a unit.

(\Leftarrow) Suppose a = ub where u \in R is a unit. Then we have (a) = aR = ubR = buR = bR = (b); note that uR = R by Proposition 9 in the text.

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