Exercise 4, § 10

Let $\mathbf{A}$ be a ring, $\mathbf{C}$ its centre, $\mathbf{B}$ the ring of matrices $\mathbf{M}_n(\mathbf{A})$, $\Delta$ the additive subgroup of $\mathbf{A}$ generated by the elements $\alpha \beta - \beta \alpha$ for $\alpha \in \mathbf{A},\ \beta \in \mathbf{A}$, and $\mathbf{D}$ the additive subgroup of $\mathbf{B}$ generated by the matrices $XY - YX$ for $X \in \mathbf{B},\ Y \in \mathbf{B}$; $\Delta$ and $\mathbf{D}$ are $\mathbf{C}$-modules. For every matrix $X = (\xi_{ij}) \in \mathbf{B}$, let $\theta(X)$ be the element $\sum_{i=1}^n \bar{\xi}_{ij}$ of $\mathbf{A}/\Delta$, where, for all $\alpha \in \mathbf{A}$, $\bar{\alpha}$ denotes the class of $\alpha$ mod. $\Delta$. Show that $\theta$ is a surjective homomorphism of $\mathbf{B}$ onto $\mathbf{A}/\Delta$, whose kernel is equal to $\mathbf{D}$, so that $\mathbf{B}/\mathbf{D}$ is isomorphic to $\mathbf{A}/\Delta$ as a $\mathbf{C}$-module. (Observe that $\mathbf{D}$ contains the matrices $\alpha E_{ij}$ and the matrices $\alpha(E_{ii} - E_{jj})$ for $\alpha \in \mathbf{A}$ and $i \neq j$.)