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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Summer 2016 • Question 4

Question

Math 6310 • Algebra Summer 2016 • Question 4

Let \(R=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \(R^2\longrightarrow R^3\) given by the matrix \[ \begin {pmatrix} x & 0\\ x & x^2\\ 1 & 1 \end {pmatrix}. \] Write \(M\) as a direct sum of cyclic \(R\)-modules.

Raw LaTeX

Let \(R=\mathbb{Q}[x]\), and let \(M\) be the cokernel of the map
\(R^2\longrightarrow R^3\) given by the matrix
\[
\begin{pmatrix}
x & 0\\
x & x^2\\
1 & 1
\end{pmatrix}.
\]
Write \(M\) as a direct sum of cyclic \(R\)-modules.

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Abelian Groups and Direct Sum Decompositions Cyclic Module Decompositions Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization