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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Summer 2020 • Question 4

Question

Math 6310 • Algebra Summer 2020 • Question 4

Suppose \(R=\mathbb {F}_3[x]\). Let \(M\) be the module generated by three elements \(a,b,c\) subject to the three relations \(-xa+x^2b+(x^2-1)c=0\), \(xb+xc=0\) and \(xb+x^2c=0\). Find a direct sum of cyclic \(R\)-modules isomorphic to \(M\).

Raw LaTeX

Suppose \(R=\mathbb{F}_3[x]\). Let \(M\) be the module generated by three elements \(a,b,c\)
subject to the three relations
\(-xa+x^2b+(x^2-1)c=0\), \(xb+xc=0\) and \(xb+x^2c=0\).
Find a direct sum of cyclic \(R\)-modules isomorphic to \(M\).

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Abelian Groups and Direct Sum Decompositions Cyclic Module Decompositions Finite Fields