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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2004 • Question 7

Question

Math 6310 • Algebra Spring 2004 • Question 7

Let \(R\) be a ring and let \(L\) be the set of all nilpotent elements of \(R\).

  1. If \(R\) is commutative, prove that \(L\) is an ideal.
  2. Give an example of a ring \(R\) in which \(L\) is not an ideal.

Raw LaTeX

Let \(R\) be a ring and let \(L\) be the set of all nilpotent elements of \(R\).
\begin{enumerate}
\item[(a)] If \(R\) is commutative, prove that \(L\) is an ideal.
\item[(b)] Give an example of a ring \(R\) in which \(L\) is not an ideal.
\end{enumerate}

Topic Tags

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