If you find any mistakes, please make a comment! Thank you.

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 is also an integral domain.


Solution: Let α and β be two nonzero elements in R[[x]]. Let α=n0anxn,β=n0bnxn.Suppose i is the smallest nonnegative integer n such that an0 and j is the smallest nonnegative integer m such that bm0. Then ai0, bj0 and α=nianxn,β=mjbmxm.Then it is clear that αβ=aibjxi+j+terms with higher degree.Because R is an integral domain, we have aibj0. So αβ0. Therefore the ring R[[x]] of formal power series is also an integral domain.


Linearity

This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.
Close Menu
Close Menu