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\title{Math 726\\Summer 2005\\Homework 3}
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\centerline{\it Due Monday July 18, 2005.}

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\noindent 1. (5pt) If $F$ is an additive, right exact functor that preserves sums, show that $F$ preserves direct limits.

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\noindent 2. (5 pt) Prove that any $R-$module $A$ is the direct limit of its finitely generated submodules.

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\noindent 3. (5 pt) Let $A_1\subseteq A_2\subseteq A_3\subseteq\cdots$ be an ascending chain of sets. Compute $\underrightarrow{\lim} A_i$.

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\noindent 4. (5 pt) Prove that the inverse limit, if it exists, is unique. Then show that if the index set has the trivial quasi-order, then $\underleftarrow{\lim} A_i\cong\prod A_i$.

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\noindent 5. (5 pt) Show that if $p$ is prime then the $\mathbb{Z}-$module, $\mathbb{Z}_p$, is an uncountable principal ideal domain with a unique nonzero prime ideal. Also show that if $I\subseteq\mathbb{Z}_p$ is any nonzero ideal, then $\mathbb{Z}_p/I$ is a finite ring.






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