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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2022 • Question 4

Question

Math 6310 • Algebra Spring 2022 • Question 4

Let \(R\) be a commutative ring with identity such that \(IJ = I \cap J\) for all ideals \(I\) and \(J\). Prove that each prime ideal of \(R\) is maximal.

Raw LaTeX

Let \(R\) be a commutative ring with identity such that \(IJ = I \cap J\) for all ideals \(I\) and \(J\). Prove that each prime ideal
of \(R\) is maximal.

Topic Tags

Ideals and Quotient Rings Prime and Maximal Ideals