Proposition 2

a) The center of a semisimple ring is semisimple.

b) The opposite ring of a semisimple ring is semisimple.

c) The quotient of a semisimple ring by a two-sided ideal is a semisimple ring.

d) The product of a finite family of semisimple rings is a semisimple ring.

Let A be a semisimple ring. It is isomorphic to the product of a finite family $(A_i)_{i\in I}$ of simple rings. The center of A is isomorphic to the product of the centers of the $A_i$, and $A^o$ is isomorphic to the product ring of the $A^o_i$. Assertions a) and b) therefore follow from Corollary 1 of VIII, p. 121.

Let $\mathfrak{a}$ be a two-sided ideal of A. The A-module $A_s/\mathfrak{a}$, quotient of the semisimple A-module $A_s$, is semisimple. The $(A/\mathfrak{a}$)-module $A_s/\mathfrak{a}$ is therefore semisimple; assertion c) follows.

A ring is semisimple if and only if it is isomorphic to the product of a finite family of simple rings; assertion d) follows.