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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2019 • Question 6

Question

Math 6310 • Algebra Fall 2019 • Question 6

Let \(R=\mathbb {F}_3[x]\). Consider the \(3\times 3\) matrix \[ \begin {bmatrix} 0 & x & x\\ 1 & x & 1+x\\ 1+x^2 & x^2 & 1+x^2 \end {bmatrix} \] Let \(\varphi :R^3\longrightarrow R^3\) be the \(R\)-module homomorphism given by this matrix. Write the cokernel of \(\varphi \) as a direct sum of cyclic modules.

Raw LaTeX

Let \(R=\mathbb{F}_3[x]\). Consider the \(3\times 3\) matrix
\[
\begin{bmatrix}
0 & x & x\\
1 & x & 1+x\\
1+x^2 & x^2 & 1+x^2
\end{bmatrix}
\]
Let \(\varphi:R^3\longrightarrow R^3\) be the \(R\)-module homomorphism given by this matrix. Write
the cokernel of \(\varphi\) as a direct sum of cyclic modules.

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Abelian Groups and Direct Sum Decompositions Cyclic Module Decompositions Hom Functors and Module Homomorphisms