Answer to Question #109070 in Abstract Algebra for Rohit

Question

Which of the following statements are true? Give reasons for your answers, in the form
of a short proof or a counterexample.
i) ( ) M3 Z has no nilpotent elements.

Answer 1

The statement is False.

Counterexample:

M3[Z] = $\begin{pmatrix} 5&-3&2\\15&-9&6\\10&-6&4 \end{pmatrix}$ is a nilpotent element because Z²=O.

O=$\begin{pmatrix} 0&0&0\\0&0&0\\0&0&0 \end{pmatrix}$