In an integral domain, there are at most two square roots of 1
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.11 Prove that if $R$ is an integral domain and $x^2 = 1$ for some $x \in…
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…
Compute the center of the Hamiltonian quaternions
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.8 Describe $Z(\mathbb{H}), where \mathbb{H}$ denotes the Hamiltonian Quaternions. Prove that $\{a+bi \ |\ a,b \in \mathbb{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…
Decide whether a given set of rationals is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.5 Decide which of the following are subrings of $\mathbb{Q}$. (1) The set of all rational numbers…
The intersection of a nonempty collection of subrings is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.4 Prove that the intersection of any nonempty collection of subrings of a ring is also a…
In a subring containing the identity, units are units in the ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.3 Let $R$ be a ring with identity and let $S \subseteq R$ be a subring containing…
In a unital ring, the negative of a unit is a unit
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.2 Let $R$ be a ring with 1. Prove that if $u$ is a unit in $R$,…