The set of ideals of a ring is closed under arbitrary intersections
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.18 Solution: (1) In Exercise 7.1.4, we showed that $I \cap J$ is a subring of $R$.…
The union of a chain of ideals is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.19 Solution: In Exercise 2.1.15, we saw that $S \subseteq R$ is an additive subgroup. To show…