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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2013 • Question 4

Question

Math 6310 • Algebra Spring 2013 • Question 4

Let \(R\) be a commutative ring with \(1\ne 0\). Recall that the nilradical of \(R\) is the ideal \[ \mathcal {N}(R)=\{x\in R:x^n=0\text { for some positive integer }n\}. \] Prove that the following are equivalent:

  1. \(R\) has exactly one prime ideal;
  2. \(R/\mathcal {N}(R)\) is a field.

Raw LaTeX

Let \(R\) be a commutative ring with \(1\ne 0\). Recall that the nilradical of \(R\) is the ideal
\[
\mathcal{N}(R)=\{x\in R:x^n=0\text{ for some positive integer }n\}.
\]
Prove that the following are equivalent:
\begin{enumerate}
\item[(i)] \(R\) has exactly one prime ideal;
\item[(ii)] \(R/\mathcal{N}(R)\) is a field.
\end{enumerate}

Topic Tags

Ideals and Quotient Rings Ideals in Quadratic Integer Rings and Exterior Powers Prime and Maximal Ideals Radical Ideals and Nilradicals