## In a commutative ring, prime ideals are radical

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.31 Solution: We begin with a lemma. Lemma: Let $R$ be a commutative ring and let $I…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.30 Solution: First, we certainly have $I \subseteq \mathsf{rad}(I)$ since for all $a \in I$, $a^1 \in…