Question

Let R = Mn(k) where k is a ring, and let A be the right ideal of R consisting of matrices whose first r rows are zero. Compute the idealizer IR(A) and the eigenring ER(A).

Answer 1

Write a matrix $\beta \in R$ in the block form $\left( \begin{array}{cc}x & y\\ z & w \end{array} \right)$, where $x \in \mathrm{Mr}(k)$, and similarly, write $\alpha \in A$ in the block form $\alpha = \left( \begin{array}{cc}0 & 0\\ u & v \end{array} \right)$. Since $\beta \alpha = \left( \begin{array}{cc}x & y\\ z & w \end{array} \right)\left( \begin{array}{cc}0 & 0\\ u & v \end{array} \right) = \left( \begin{array}{cc}yu & yv\\ wu & wv \end{array} \right)$, the condition for $\beta A \subseteq A$ amounts to $y = 0$. Therefore, $\operatorname{IR}(A)$ is given by

the ring of "block lower-triangular" matrices $\left\{\left( \begin{array}{cc}x & 0\\ z & w \end{array} \right)\right\}$. Quotienting out the ideal $\left\{\left( \begin{array}{cc}0 & 0\\ z & w \end{array} \right)\right\}$, we get the eigenring $\operatorname{ER}(A) \sim \operatorname{Mr}(k)$.