A fact about ideals and subrings which intersect trivially
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.22 Solution: By the Second Isomorphism Theorem for rings, we have $$(S+I)/I \cong S/(S \cap I) =…
The intersection of an ideal and a subring is a subideal, but not necessarily vice versa
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.20 Solution: $I \cap S$ is a subring by Exercise 7.1.4, so it suffices to show absorption.…
Decide whether or not a subset of Z[x] is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.10 Solution: (1) We claim that this subset $S$ is an ideal. To that end, suppose $\alpha…
Determine whether a subring is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.9 Solution: We have already seen which of these are subrings. (1) Let $S = \{ f…
The diagonal subset of the Cartesian square of a ring is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.18 Let $R$ be a ring. Prove that $\Delta(R) = \{(r,r) \ |\ r \in R \}$…
Every subring of a field which contains 1 is an integral domain
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.12 Prove that any subring of a field which contains the identity is an integral domain. Solution:…
In a division ring, every centralizer is a division ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.10 Prove that if $D$ is a division ring, then $C_D(a)$ is a division ring for all…
Basic properties of centralizers in a ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.9 Let $R$ be a ring. For a fixed element $a \in R$, define $C_R(a) = \{r…
The center of a ring is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.7 The center of a ring $R$ is $$Z(R) = \{ z \in R \ |\ zr…
Decide whether a given subset is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.6 Decide which of the following are subrings of the ring of all functions from the closed…
- 1
- 2