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Abstract Algebra - Part II

Standard Form

Problems 16-19 of Section 12.1 work through the reduction algorithm to standard form in the slightly simpler, but practically useful case of $R$ being a Euclidean Ring.

Torsion and Rank

Remember that $\Tor(M)=\lbrace m\in M : \exists r\in R, rm=0\rbrace$ and that the rank of a module is the maximal size of a linearly independent set of elements of $M$.

  1. (DF Section 12.1 problem 1) Let $M$ be a module over an integral domain $R$. Show that the rank of $M$ (the maximal number of linearly independent elements of $M$) is zero. Then show that the rank of $M$ is the same as the rank of $M/\Tor(M)$.

More on Torsion and Rank

  1. (DF Section 12.1 problem 2) Let $M$ be a module over an integral domain $R$.

    a. Let $x_1,\ldots, x_n$ be a maximal linearly independent set of elements of $M$. Let $N$ be the submodule of $M$ generated by the $x_{i}$. Prove that $N$ is isomorphic to $R^{n}$ and that $M/N$ is a torsion $R$-module.

    b. Conversely, suppose $M$ contains a free $R$-module $N$ of rank $n$ such that $M/N$ is torsion. Prove that $M$ has rank $n$.

  2. (DF Section 12.1 Problem 4) So that if $M$ has rank $n$, $N\subset M$ has rank $r$, and $M/N$ has rank $s$, then $n=r+s$.

Primary Decomposition

  1. (DF Section 12.1 problem 11) Let $R$ be a PID and let $a\in R$ be a nonzero element. Let $M=R/aR$. For any prime $p$ of $R$, prove that, if $n$ is the power of $p$ dividing $a$ in $R$, then
\[p^{k-1}M/p^{k}M=\begin{cases} R/pR & k\le n \\ 0 & k>n\end{cases}\]

More on primary decomposition (uniqueness part of main theorem)

  1. (DF Section 12.1 problem 12) Let $R$ be a PID and let $p$ be a prime in $R$. This problem gives an approach to proving uniqueness of the decomposition into cyclic modules of the form $R/p^{a}R$.

    a. Let $M$ be a finitely generated torsion $R$-module. Prove that $p^{k-1}M/p^{k}M$ is isomorphic to $F^{n_{k}}$ where $F$ is the field $R/pR$ and $n_{k}$ is the number of elementary divisors of $M$ which are powers of $p^{\alpha}$ with $\alpha\ge k$.

    b. Suppose $M_1$ and $M_2$ are isomorphic finitely generated torsion $R$-modules. Use part (a) to prove that, for all $k\ge 0$, $M_1$ and $M_2$ have the same number of elementary divisors $p^{\alpha}$ with $\alpha\ge k$. Show that this implies that $M_1$ and $M_2$ have the same set of elementary divisors.