Algebra Qual 2020-09

Problem 1

Let $A=\left[\begin{array}{ccc}6& -2& -1\\ 10& -3& -2\\ 0& 0& 1\end{array}\right]$.
(a) Find bases for the eigenspaces of $A$.
(b) Determine if $A$ is diagonalizable. If so, give an invertible matrix $P$ and diagonal matrix $D$ such that $P^{-1}AP=D$. If not, explain why not.

Problem 2

Let $G$ be the additive group ${\mathbf{Z}}_{2020}$ and let $H\subseteq G$ be the subset consisting of those elements with order dividing 20.
(a) Prove $H$ is a subgroup of $G$.
(b) Find an explicit generator for $H$ and determine its order.

Problem 3

Let $G$ be a finite group and $\mathrm{Z}(G)$ denote its center.
(a) Prove that if $G/\mathrm{Z}(G)$ is cyclic, then $G$ is abelian.
(b) Prove that if $G$ is nonabelian, then $|\mathrm{Z}(G)|\le \frac{1}{4}|G|$.

Problem 4

Let $R$ be a commutative ring with 1. We say an element $n\in R$ is nilpotent if there exists a number $k\in \mathbf{N}$ such that $n^k=0$.
(a) Show that if $n$ is nilpotent, then $1-n$ is a unit.
(b) Give an example of a commutative ring with 1 that has no nonzero nilpotent elements, but is not an integral domain.

Problem 5

Let $R$ be a commutative ring with 1 and suppose $e\in R$ is idempotent, i.e., satisfies $e^2=e$.
(a) Prove that $1-e$ is also idempotent.
(b) Suppose $e\ne 0,1$. Show that $Re$ and $R(1-e)$ are proper ideals of $R$.
(c) Prove there is an isomorphism $R\cong Re\times R(1-e)$.