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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…

The ring of formal power series over an integral domain is an integral domain

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.4 Prove that if $R$ is an integral domain then the ring $R[[x]]$ of formal power series…

The set of formal power series is a ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.3 Let $R$ be a ring. Define the set $R[[x]]$ of formal power series in the indeterminate…