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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2022 • Question 2

Question

Math 6310 • Algebra Fall 2022 • Question 2

Let \(k=\mathbb {Z}/2\mathbb {Z}\) be the field with two elements and let \(R=k[x]\). Let \(M\) be the cokernel of the mapping from \(R^3\) to \(R^3\) given by the matrix \[ \begin {bmatrix} x^2&x^2&0\\ x^2&x^2+x&x+1\\ 0&0&x+1 \end {bmatrix} \]

How many elements are in \(\operatorname {Hom}_R(k[x]/(x),M)\)?

Raw LaTeX

Let \(k=\mathbb{Z}/2\mathbb{Z}\) be the field with two elements and let \(R=k[x]\). Let \(M\) be the cokernel of the mapping from \(R^3\) to
\(R^3\) given by the matrix
\[
\begin{bmatrix}
x^2&x^2&0\\
x^2&x^2+x&x+1\\
0&0&x+1
\end{bmatrix}
\]

How many elements are in \(\operatorname{Hom}_R(k[x]/(x),M)\)?

Topic Tags

Cyclic Module Decompositions Hom Functors and Module Homomorphisms Ideals and Quotient Rings Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization