Find all the invertible elements, zero dividers and nilpontent elements of the ring.

Question

The task is to find all the invertible elements, zero dividers and nilpontent elements of the ring $$R = \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} | a,b,c \in \mathbb{R} \right\}$$ with standart addition and multiplication operators.

I have no idea how to do that other than simply bruteforce, however in terms of this task that seems to be terrible. Would be grateful for any other solutions?

Answer 1

For invertible elements you are looking for matrices $A \in R$ such that there exists $M\in R$ for which $AM=MA=I=\begin{bmatrix}1&0\\0&1\end{bmatrix} \in R$. So for that to happen \begin{align*} \begin{bmatrix}a&b\\0&c\end{bmatrix}\begin{bmatrix}p&q\\0&r\end{bmatrix}=\begin{bmatrix}p&q\\0&r\end{bmatrix}\begin{bmatrix}a&b\\0&c\end{bmatrix}& =\begin{bmatrix}1&0\\0&1\end{bmatrix}\\ \begin{bmatrix}ap&aq+br\\0&cr\end{bmatrix}=\begin{bmatrix}pa&pb+qc\\0&cr\end{bmatrix}& =\begin{bmatrix}1&0\\0&1\end{bmatrix}. \end{align*} This gives us \begin{align*} ap & =1\\ cr & =1\\ aq+br=pb+qc&=0 \end{align*} From the first two equations we get that $a,c\neq 0$ So matrices $A \in R$ such that the diagonal entries are non-zero will be invertible. $$I=\left\{\begin{bmatrix}a&b\\0&c\end{bmatrix} \, \bigg | \, a,c \neq 0\right\}.$$ Hopefully you can take the rest from here.

Answer 2

Hint for invertible elements: they would have to be nonsingular matrices, in particular. Do you know how to use the determinant to detect nonsingular matrices?

Hint for zero divisors: it is not hard to show any singular matrix in this ring is a zero divisor. Since nonsingular matrices are invertible, you will then have exactly identified the zero divisors.

Hint for nilpotent elements: for a general element, look at what happens on the diagonal when you raise the matrix to powers. You will be able to conclude something about the diagonal, and then you have your anwswer.