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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2009 • Question 5

Question

Math 6310 • Algebra Spring 2009 • Question 5

Let \(R\) be a commutative ring. If for each prime ideal \(\mathfrak {p}\) of \(R\), the local ring \(R_{\mathfrak {p}}\) contains no nonzero nilpotent elements, prove that \(R\) contains no nonzero nilpotent elements.

Raw LaTeX

Let \(R\) be a commutative ring. If for each prime ideal \(\mathfrak{p}\) of \(R\), the local ring
\(R_{\mathfrak{p}}\) contains no nonzero nilpotent elements, prove that \(R\) contains no nonzero
nilpotent elements.

Topic Tags

Ideals and Quotient Rings Polynomial Rings and Prime Ideals Prime and Maximal Ideals Radical Ideals and Nilradicals