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

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.8

Solution:

($\Rightarrow$) 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.