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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2021 • Question 2

Question

Math 6310 • Algebra Fall 2021 • Question 2

Let \(R:=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \[ R^3\xrightarrow {\begin {pmatrix}x-2&4&x\\0&4&x\\0&-x&-1\end {pmatrix}}R^3. \]

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
\[
R^3\xrightarrow{\begin{pmatrix}x-2&4&x\\0&4&x\\0&-x&-1\end{pmatrix}}R^3.
\]

Write \(M\) as a direct sum of cyclic \(R\)-modules.

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Cyclic Module Decompositions Polynomial Rings and Ring Homomorphism Kernels