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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2023 • Question 2

Question

Math 6310 • Algebra Fall 2023 • Question 2

Suppose \(R\) is a commutative ring with 1 and \(x, y \in R\) satisfy \(xy = 0\) and \((x, y) = R\) (i.e. the ideal generated by \(x\) and \(y\) is the whole ring). Prove that the \(R\)-module homomorphism

\[ \psi :R\to R/(x)\oplus R/(y) \]

sending \(f\in R\) to its image in \(R/(x)\), and in \(R/(y)\), is an isomorphism.

Raw LaTeX

Suppose \(R\) is a commutative ring with 1 and \(x, y \in R\) satisfy \(xy = 0\) and \((x, y) = R\) (i.e. the ideal generated by \(x\)
and \(y\) is the whole ring). Prove that the \(R\)-module homomorphism

\[
\psi:R\to R/(x)\oplus R/(y)
\]


sending \(f\in R\) to its image in \(R/(x)\), and in \(R/(y)\), is an isomorphism.

Topic Tags

Hom Functors and Module Homomorphisms Ideals and Quotient Rings Modules over PIDs Polynomial Rings and Ring Homomorphism Kernels