Characterize the two-sided ideals of a matrix ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.21 Solution: First, let $I \subseteq M_n(R)$ be a two-sided ideal. Let $J \subseteq R$ consist of…
The intersection of an ideal and a subring is a subideal, but not necessarily vice versa
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.20 Solution: $I \cap S$ is a subring by Exercise 7.1.4, so it suffices to show absorption.…
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…
Decide whether or not a subset of Z[x] is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.10 Solution: (1) We claim that this subset $S$ is an ideal. To that end, suppose $\alpha…
Determine whether a subring is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.9 Solution: We have already seen which of these are subrings. (1) Let $S = \{ f…
Determine whether or not a subset is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.8 Solution: (1) Note that $(1,1) \in D$. However, $(1,0)(1,1) = (1,0) \notin D$. Since $D$ does…
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