Example

Let K be an algebraically closed commutative field, and let A be a semisimple algebra of finite degree over K. Let $(V_i)_{i\in I}$ be a family of simple A-modules such that every simple A-module is isomorphic to a unique $V_i$. Then I is a finite set (VIII, p. 136, Proposition 1), the vector spaces $V_i$ are finite-dimensional over the field K, the commutant of $V_i$ is equal to $K\cdot 1_{V_i}$ (VIII, p. 47, Theorem 1), and the mapping $a\mapsto (a_{V_i})_{i\in I}$ is an algebra isomorphism from A to $\prod_{i\in I}$ End$_K(V_i)$ (Proposition 8).

We have defined (VIII, p. 50) a minimal two-sided ideal as a minimal element of the set of nonzero two-sided ideals, ordered by inclusion. In other words, a minimal two-sided ideal $\mathfrak{a}$ of A is a simple $(A$, A)-sub-bimodule of $_sA_d$. Likewise, we define a maximal two-sided ideal of A as a maximal element of the set of proper two-sided ideals of A. A maximal two-sided ideal $\mathfrak{a}$ of A is simply a maximal $(A$, A)-sub-bimodule of $_sA_d$ (VIII, p. 48, Definition 2). If the ring A is simple, then the ideal 0 is a maximal two-sided ideal of A and A is a minimal two-sided ideal.

For any $\lambda \in \mathscr{S}$, we denote the isotypical component of type $\lambda$ of the A-module $A_s$ by $\mathfrak{a}_{\lambda}$. For any subset Λ of $\mathscr{S}$, we set $\mathfrak{a}_{\Lambda}=\sum_{\lambda\in\Lambda}\mathfrak{a}_{\lambda}$.