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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2018 • Question 6

Question

Math 6310 • Algebra Fall 2018 • Question 6

Let \(A=\mathbb {F}_3[x]\), i.e., \(A\) is a polynomial ring in one variable over the field with 3 elements. Suppose \(M\) and \(N\) are finitely generated \(A\)-modules such that \[ M\oplus \frac {A}{x^3+1} \cong N\oplus \frac {A}{x+1}\oplus \frac {A}{x+1}\oplus \frac {A}{x+1}. \] Are \(M\) and \(N\) isomorphic?

Raw LaTeX

Let \(A=\mathbb{F}_3[x]\), i.e., \(A\) is a polynomial ring in one variable over the field with
3 elements. Suppose \(M\) and \(N\) are finitely generated \(A\)-modules such that
\[
M\oplus \frac{A}{x^3+1}
\cong
N\oplus \frac{A}{x+1}\oplus \frac{A}{x+1}\oplus \frac{A}{x+1}.
\]
Are \(M\) and \(N\) isomorphic?

Topic Tags

Field Extensions Finite Fields Polynomial Rings and Prime Ideals Polynomial Rings and Ring Homomorphism Kernels