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\name{}
%let me know if you have a better way to do this (underline) that doesn't require the ulem package.
\class{Math 171 - Abstract Algebra I Section \_\!\_\!\_\!\_}
\assignment{HW \#10}
\duedate{10/11/10}

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\begin{document}

\problemlist{Dummit \& Foote (7.2) 3,7,10,13}

\begin{problem}[7.2.3]
Let $R$ be a commutative ring with identity and define the set $R[[x]]$ of \textit{formal power series} in $x$ with coefficients from $R$ to be all formal infinite sums \[ \sum_{n=0}^{\infty} a_{n}x^{n} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots .\] Recall that addition and multiplication are defined in essentially the same way as for polynomials. \[ \left(\sum_{n=0}^{\infty} a_{n}x^{n}\right) + \left(\sum_{n=0}^{\infty} b_{n}x^{n}\right) = \sum_{n=0}^{\infty} (a_{n}+b_{n})x^{n} \]
\[ \left( \sum_{n=0}^{\infty} a_{n}x^{n} \right) \times \left( \sum_{n=0}^{\infty} b_{n}x^{n} \right) = \sum_{n=0}^{\infty} \left(\sum_{k=0}^{n} a_{k}b_{n-k}\right)x^{n}\]
\begin{itemize}
\item[(a)] Prove that $R[[x]]$ is a commutative ring with identity.
\item[(b)] Show that $1-x$ is a unit in $R[[x]]$ with inverse $1+x+x^{2} + \ldots$.
\item[(c)] Prove that $\sum_{n=0}^{\infty} a_{n}x^{n}$ is a unit in $R[[x]]$ if and only if $a_{0}$ is a unit in $R$.
\end{itemize}
\end{problem}
\newpage


\begin{problem}[7.2.7]
%this should be R = ring, not \mathbb{R} = real numbers, right?
The center of a ring $R$ is the set \[ Z(R) = \{ r \in R \, | \, rx = xr \textrm{ for all } x \in R \}.\] Let $R$ be a commutative ring with identity. Prove that the center of the ring $M_n(R)$ is the set of scalar matrices, which are scalar multiples of the identity matrix. [Use the elements $E_{ij}$ that we talked about in class.]
\end{problem}
\newpage

\begin{problem}[7.2.10]
Consider the following elements of the integral group ring $\Z S_3$:
$$\alpha = 3(1\ 2)-5(2\ 3)+14(1\ 2\ 3)\quad \text{and} \quad \beta = 6(1)+2(2\ 3)-7(1\ 3\ 2)$$
(where (1) is the identity of $S_3$).  Compute the following elements:
\begin{itemize}
\item[(a)] $\alpha + \beta$,
\item[(b)] $2\alpha - 3\beta$,
\item[(c)] $\alpha \beta$,
\item[(d)] $\beta \alpha$,
\item[(e)] $\alpha^2$.
\end{itemize}
\end{problem}
\newpage

\begin{problem}[7.2.13]
Assume $R$ is a commutative ring with identity. Let $\mathcal{K} = \{k_1,\ldots,k_m\}$ be a conjugacy class in the finite group $G$.
\begin{itemize}
\item[(a)] Prove that the element $K = k_1 + \ldots + k_m$ is in the center of the group ring $RG$ (cf. Exercise 7, Section 1).  [Check that $g^{-1}Kg = K$ for all $g \in G$.]
\item[(b)] Let $\mathcal{K}_1,\ldots,\mathcal{K}_r$ be the conjugacy classes of $G$ and for each $\mathcal{K}_i$ let $K_i$ be the element of $RG$ that is the sum of the members of $\mathcal{K}_i$.  Prove that an element $\alpha \in RG$ is in the center of $RG$ if and only if $\alpha = a_1K_1 + a_2K_2 + \ldots + a_rK_r$ for some $a_1,a_2,\ldots,a_r \in R$.
\end{itemize}

\end{problem}

\end{document}