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## The ideal generated by the variable is maximal iff the coefficient ring is a field

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.18 Solution: We first prove a lemma. Lemma: The map $\varphi : R[[x]] \rightarrow R$ given by…

## 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 $a = ub$…