Corollary

a) The minimal left ideals of A are the ideals $\mathfrak{a}(H)$, where H is a hyperplane in V. The maximal left ideals of A are the ideals $\mathfrak{a}(L)$, where L is a line in V.

b) The minimal right ideals of A are the ideals $\mathfrak{b}(L)$, where L is a line in V. The maximal right ideals of A are the ideals $\mathfrak{b}(H)$, where H is a hyperplane in V.

Let $(L_i)_{i\in I}$ be a family of lines with direct sum V. Let $(\varepsilon_i)_{i\in I}$ be the family of projectors associated with the decomposition $V =\oplus_{i\in I}L_i$. The $\varepsilon_i$ are idempotent in A, and we have $\varepsilon_i\varepsilon_j= 0$ for $i\not=j$ and $\sum_{i\in I}\varepsilon_i= 1$. Denote the hyperplane $\sum_{j\not=i}L_j$ by $H_i$; it is the kernel of $\varepsilon_i$. We then have

The A-module $A_s$ is the direct sum of the family $(\mathfrak{a}(H_i))_{i\in I}$ of minimal left ideals, and $A_d$ is the direct sum of the family $(\mathfrak{b}(L_i))_{i\in I}$ of minimal right ideals.

Consider the specific case $V = D^n_d$, and identify A with the matrix ring $\mathbf{M}_n(D)$. Denote the interval $[1, n]$ in $\mathbf{N}$ by I and the canonical basis of V by $(v_i)_{i\in I}$; set $L_i=v_iD$, and denote by $E_{ij}$ the matrix units (II, §10, No. 3, p. 341). We then have $\varepsilon_i= E_{ii}$. The left ideal AE$_{ii}$ is equal to DE$_{1i}+\cdots +$ DE$_{ni}$ and consists of the matrices with all columns except the $i$-th equal to zero. The right ideal $E_{ii}A$ is equal to DE$_{i1}+\cdots +$ DE$_{in}$ and consists of the matrices with all lines except the $i$th equal to zero. We also have the relation

for $i$ and $j$ between 1 and $n$.