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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 φ:R[[x]]R given by rixir0 is a surjective ring homomorphism and ker φ=(x).

Proof: φ is clearly a surjective homomorphism. Now suppose rixiker φ. Then r0=0, and rixi=xri+1xi(x). If α=xaixi, then φ(α)=0, so that αker φ. ◻

By the First Isomorphism Theorem for rings, R[[x]]/(x)R. The problem at hand then follows from Propositions 12 and 13 in the text.


Linearity

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