## 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…

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…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.15 Prove that a quotient of a divisible abelian group by any proper subgroup is also divisible.…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.5 Prove for all $n > 1$ that $\mathbb{Z}/(n)$ is not a group under multiplication of residue…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.4 Prove that multiplication of residue classes in $\mathbb{Z}/(n)$ is associative. (You may assume that it is…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.3 Prove that addition of residue classes in $\mathbb{Z}/(n)$ is associative. (You may assume that it is…