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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2007 • Question 3

Question

Math 6310 • Algebra Fall 2007 • Question 3

Let \(M\) be a \(\mathbb {Z}\)-module consisting of \(3\times 1\)-matrics with integer entries. Let \(N\subseteq M\) be a \(\mathbb {Z}\)-submodule generated by the columns of the matrix \[ \begin {pmatrix} 2 & 7 & 3 & 4\\ 3 & 8 & 7 & 11\\ 4 & 10 & 10 & 16 \end {pmatrix}. \] Find a minimal presentation for the quotient \(M/N\).

Raw LaTeX

Let \(M\) be a \(\mathbb{Z}\)-module consisting of \(3\times 1\)-matrics with integer entries.
Let \(N\subseteq M\) be a \(\mathbb{Z}\)-submodule generated by the columns of the matrix
\[
\begin{pmatrix}
2 & 7 & 3 & 4\\
3 & 8 & 7 & 11\\
4 & 10 & 10 & 16
\end{pmatrix}.
\]
Find a minimal presentation for the quotient \(M/N\).

Topic Tags

Free Groups and Group Presentations Ideals and Quotient Rings Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization Modules over PIDs