Exercise 31, § 9

Let K be a commutative field, and let A be the set of matrices of the form $\begin{pmatrix} a & b \\ 0 & c \end{pmatrix}$ with $a, b, c\in K$. Verify that A is a subring of the matrix ring and that its radical is determined by $a=c= 0$. Prove that the image of $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ in $A/\mathfrak{R}(A)$ is an idempotent element of the center of $A/\mathfrak{R}(A)$.