Let \(R\) denote a commutative ring and \(I\) an ideal, \(I \neq R\). We say that \(R\) has nilpotents if there exists \(r \in R\), \(r \neq 0\), with \(r^n = 0\).
Give an example where \(R/I\) has nilpotents but \(R\) doesn’t.
Give an example where \(R\) has nilpotents but \(R/I\) doesn’t.
Solution
Part (a).
1. Take \(R = \ZZ\) and \(I = 4\ZZ\).
\(\ZZ\) is a domain, so it has no nonzero nilpotents.
2. \(R/I = \ZZ/4\ZZ\) has a nilpotent.
the class \(\overline{2}\) satisfies \(\overline{2} \neq 0\) but \(\overline{2}^2 = \overline{4} = \overline{0}\).
3. Hence \(R/I\) has nilpotents while \(R\) does not.
1 and 2.
Part (b).
1. Take \(R = \ZZ/4\ZZ\) and \(I = 2\ZZ/4\ZZ\).
\(R\) has the nilpotent \(\overline{2}\).
2. \(R/I \cong \ZZ/2\ZZ\).
the quotient of \(\ZZ/4\ZZ\) by the ideal generated by \(\overline{2}\) is \(\ZZ/2\ZZ\).
3. \(\ZZ/2\ZZ\) has no nonzero nilpotents.
it is a field, hence a domain.
4. Hence \(R\) has nilpotents while \(R/I\) does not.
1 and 3.
5. Q.E.D.
3 (part (a)) and 4 (part (b)).