Corollary 1

a) The center of a simple ring is a field.

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

c) The left length of a simple ring is equal to its right length.

Let D be a field, Z its center, and V a right vector space over D of finite dimension $r\geqslant 1$. We denote the endomorphism ring of V by E.

The mapping $z\mapsto z_V$ is an isomorphism from Z to the center of E by Corollary 2 of VIII, p. 83. Assertion a) follows. The dual $V^*$ of V is a right vector space over the opposite field $D^o$ of D, and its dimension is equal to $r$. The mapping $u\mapsto^tu$ is an isomorphism from the opposite ring $E^o$ of E to the ring End$_{D^o}(V^*)$. Consequently, the ring $E^o$ is simple, and the rings E and $E^o$ have the same left length, equal to $r$ (Lemma 2).