Basic properties of nilpotent ring elements
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.14 Let $R$ be a commutative ring and let $x \in R$ be nilpotent – that is,…
Examples of nilpotent elements
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.13 An element $x \in R$, $R$ a ring, is called nilpotent if $x^m = 0$ for some…
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 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…