Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.4
Prove that if is an integral domain then the ring of formal power series is also an integral domain.
Solution: Let and be two nonzero elements in . Let Suppose is the smallest nonnegative integer such that and is the smallest nonnegative integer such that . Then , and Then it is clear that Because is an integral domain, we have . So . Therefore the ring of formal power series is also an integral domain.