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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2011 • Question 8

Question

Math 6310 • Algebra Spring 2011 • Question 8

For a ring \(R\) define its nilradical \(n(R)=\{x\in R:x^n=0\text { for some }n\in \mathbb {Z}\}\).

  1. If \(R\) is commutative, prove that \(n(R)\) is an ideal of \(R\).
  2. Is the nilradical an ideal even if \(R\) is noncommutative?

Raw LaTeX

For a ring \(R\) define its nilradical \(n(R)=\{x\in R:x^n=0\text{ for some }n\in\mathbb{Z}\}\).
\begin{enumerate}
\item If \(R\) is commutative, prove that \(n(R)\) is an ideal of \(R\).
\item Is the nilradical an ideal even if \(R\) is noncommutative?
\end{enumerate}

Topic Tags

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