Algebra Qual 2015-03

Problem 1

Let $G$ and $H$ be groups of order 10 and 15, respectively. Prove that if there is a nontrivial homomorphism $ϕ : G \to H$ , then $G$ is abelian.

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Let $G$ and $H$ be groups of order 10 and 15, respectively. Prove that if there is a nontrivial homomorphism $\phi:G\to H$, then $G$ is abelian.

Problem 2

Let $G$ be an abelian group and $G_{T}$ be the set of elements of finite order in $G$ .

Prove that $G_{T}$ is a subgroup of $G$ .
Prove that every non-identity element of $G / G_{T}$ has infinite order.
Characterize the elements of $G_{T}$ when $G = R / Z$ , where $R$ is the additive group of real numbers.

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Let $G$ be an abelian group and $G_T$ be the set of elements of finite order in $G$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that $G_T$ is a subgroup of $G$.
	\item Prove that every non-identity element of $G/G_T$ has infinite order.
	\item Characterize the elements of $G_T$ when $G={\bf R}/{\bf Z}$, where ${\bf R}$ is the additive group of real numbers.
\end{enumerate}

Problem 3

Suppose $I$ and $J$ are ideals in a commutative ring $R$ such that $R = I + J$ . Prove that the map $f : R \to R / I \times R / J$ given by $f (x) = (x + I, x + J)$ induces the isomorphism

$R / I J ≅ R / I \times R / J .$
Prove that $(Z / 3 Z) [x] / (x^{3} - x^{2} - 1) ≅ (Z / 3 Z) [x] / (x^{3} + x + 1)$ . (Hint: Use part (1).)

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\begin{enumerate}
	\item Suppose $I$ and $J$ are ideals in a commutative ring $R$ such that $R=I+J$. Prove that the map $f:R\to R/I\times R/J$ given by $f(x)=(x+I,x+J)$ induces the isomorphism
	\[
		R/IJ\cong R/I\times R/J.
	\]
	\item Prove that $\left({\bf Z}/3{\bf Z}\right)[x]/(x^3-x^2-1)\cong \left({\bf Z}/3{\bf Z}\right)[x]/(x^3+x+1)$. ({\itshape Hint:} Use part (a).)
\end{enumerate}

Problem 4

An element $r$ of a ring $R$ is said to be idempotent if $r^{2} = r$ . Suppose that $R$ is a commutative ring with unity containing an idempotent element $e$ .

Prove that $1 - e$ is also idempotent.
Prove that $e R$ and $(1 - e) R$ are both ideals in $R$ and that

$R ≅ e R \times (1 - e) R .$
Prove that if $R$ has a unique maximal ideal, then the only idempotent elements in $R$ are 0 and 1.

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An element $r$ of a ring $R$ is said to be {\bfseries idempotent} if $r^2=r$. Suppose that $R$ is a commutative ring with unity containing an idempotent element $e$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that $1-e$ is also idempotent.
	\item Prove that $eR$ and $(1-e)R$ are both ideals in $R$ and that
	\[
		R\cong eR\times (1-e)R.
	\]
	\item Prove that if $R$ has a unique maximal ideal, then the only idempotent elements in $R$ are 0 and 1.
\end{enumerate}

Problem 5

Suppose $T$ is a linear transformation on a finite-dimensional complex inner-product space $V$ . Let $I$ denote the identity transformation on $V$ . The numerical range of $T$ is the subset of $C$ defined by

$W (T) = {⟨ T (x), x ⟩ | x \in V, ∥ x ∥ = 1} .$

Show that $W (T + c I) = W (T) + c$ for every $c \in C$ .
Show that $W (c T) = c W (T)$ for every $c \in C$ .
Show that the eigenvalues of $T$ are contained in $W (T)$ .
Let $B$ be an orthonormal basis for $V$ . Show that the diagonal entries of $[T]_{B}$ are contained in $W (T)$ .

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Suppose $T$ is a linear transformation on a finite-dimensional complex inner-product space $V$. Let $I$ denote the identity transformation on $V$. The {\bfseries numerical range} of $T$ is the subset of ${\bf C}$ defined by
\[
	\operatorname{W}(T)=\{\langle T(x),x\rangle \,|\, x\in V,\quad \|x\|=1\}.
\]

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Show that $\operatorname{W}(T+cI)=\operatorname{W}(T)+c$ for every $c\in {\bf C}$.
	\item Show that $\operatorname{W}(cT)=c\operatorname{W}(T)$ for every $c\in {\bf C}$.
	\item Show that the eigenvalues of $T$ are contained in $\operatorname{W}(T)$.
	\item Let $\mathcal{B}$ be an orthonormal basis for $V$. Show that the diagonal entries of $[T]_{\mathcal{B}}$ are contained in $\operatorname{W}(T)$.
\end{enumerate}