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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2015 • Question 6

Question

Math 6310 • Algebra Fall 2015 • Question 6

Suppose that \(R=\mathbb {Z}[i]\) is the Gaussian integers (here \(i\) is a square root of \(-1\)). Let \(M\) and \(N\) be finitely generated \(R\)-modules such that \[ M\oplus R^{\oplus 2}\oplus R/\langle 2\rangle \cong N\oplus R\oplus R/\langle 1+i\rangle \oplus R\oplus R/\langle 1-i\rangle . \] Is it true that \(M\cong N\)?

Raw LaTeX

Suppose that \(R=\mathbb{Z}[i]\) is the Gaussian integers (here \(i\) is a square root of \(-1\)).
Let \(M\) and \(N\) be finitely generated \(R\)-modules such that
\[
M\oplus R^{\oplus 2}\oplus R/\langle 2\rangle
\cong
N\oplus R\oplus R/\langle 1+i\rangle\oplus R\oplus R/\langle 1-i\rangle.
\]
Is it true that \(M\cong N\)?

Topic Tags

Ideals in Quadratic Integer Rings and Exterior Powers