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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2023 • Question 2

Question

Math 6310 • Algebra Spring 2023 • Question 2

Let \(k=\mathbb {Z}/2\mathbb {Z}\) and \(R=k[x]\). Let \(M\) denote the cokernel of the map \(R^3\to R^3\) given by the matrix: \[ \begin {bmatrix} 1+x&x^2&1+x\\ 1&x&1\\ 1&x^2&1+x^2 \end {bmatrix} \]

Write \(\operatorname {Hom}(M,R/(x))\) as a direct sum of cyclic modules.

Raw LaTeX

Let \(k=\mathbb{Z}/2\mathbb{Z}\) and \(R=k[x]\). Let \(M\) denote the cokernel of the map \(R^3\to R^3\) given by the matrix:
\[
\begin{bmatrix}
1+x&x^2&1+x\\
1&x&1\\
1&x^2&1+x^2
\end{bmatrix}
\]

Write \(\operatorname{Hom}(M,R/(x))\) as a direct sum of cyclic modules.

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Cyclic Module Decompositions Hom Functors and Module Homomorphisms Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization