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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Summer 2017 • Question 5

Question

Math 6310 • Algebra Summer 2017 • Question 5

Let \(R=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \[ R^2\longrightarrow R^3 \qquad \text {given by}\qquad \begin {pmatrix} x+1 & 0\\ 2x & x^2\\ 1 & 1 \end {pmatrix}. \] 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
\[
R^2\longrightarrow R^3
\qquad\text{given by}\qquad
\begin{pmatrix}
x+1 & 0\\
2x & x^2\\
1 & 1
\end{pmatrix}.
\]
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 Polynomial Rings and Ring Homomorphism Kernels