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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2020 • Question 5

Question

Math 6310 • Algebra Spring 2020 • Question 5

Consider \(R:=\mathbb {Q}[x]\) and the matrix: \[ A= \begin {bmatrix} x-1 & 0\\ 1-x & x^2 \end {bmatrix}. \] Let \(M\) be the cokernel of the map \(R^2\longrightarrow R^2\) given by multiplication by \(A\). Compute the rank of the \(\mathbb {Q}\)-vector space \(\operatorname {Hom}_R(M,R/(x^2))\).

Raw LaTeX

Consider \(R:=\mathbb{Q}[x]\) and the matrix:
\[
A=
\begin{bmatrix}
x-1 & 0\\
1-x & x^2
\end{bmatrix}.
\]
Let \(M\) be the cokernel of the map \(R^2\longrightarrow R^2\) given by multiplication by \(A\).
Compute the rank of the \(\mathbb{Q}\)-vector space \(\operatorname{Hom}_R(M,R/(x^2))\).

Topic Tags

Cyclic Module Decompositions Hom Functors and Module Homomorphisms Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization Polynomial Rings and Ring Homomorphism Kernels