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\name{}
\class{Math 171 - Abstract Algebra I Section \uline{\;\;}}
\assignment{HW \#22}
\duedate{12/9/10}

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\begin{problem}
1. (D \& F 10.1, 1) Let $R$ be a ring with identity and let $M$ be an $R$-module. Prove that $0 \cdot m = 0$ and $(-1) \cdot m = -m$ for all $m \in M$
\end{problem}


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\begin{problem} 2. (D \& F 10.1, 9) Let $N$ be a submodule of an $R$-module $M$. Then the annihilator of $N$ in $R$ is defined to be \[ \textrm{Ann}_{R}(N) = \{ r \in R \, | \, rn = 0 \textrm{ for all } n \in N \}.\] Show that Ann$_{R}(N)$ is an ideal of $R$.
\end{problem}

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\begin{problem}
3. Let $R = \mathbb{C}[S_{2}]$. Then the complex vector space $V$ spanned by $v_{1}$ and $v_{2}$ is an $R$-module under the following action:

\[ (ae + b(12)) \cdot (cv_{1} + dv_{2}) = (ac + bd)v_{1} + (ad +bc)v_{2},\] where $e$ denotes the identity element of $S_{2}$. (This module corresponds to the group action where $(12)$ switches $v_{1}$ and $v_{2}$.)

Find all submodules of $V$. (Hint: Argue that since $\mathbb{C}[S_{2}]$ contains elements of the form $ze$ for all $z$, a submodule of $V$ is also a subspace of $V$.)

\end{problem}


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