Two classical splitting theorems: Easy proofs

The two main splitting theorems for a division ring which is finite dimensional over its center are obtained as corollaries to:Lemma. Suppose B ? A ? D where B is a (commutative) subfield of the division ring D and A is an intermediate ring. (For any set X ? D let X′ denote the cenralizer of X in D.) If B = B″, then

$\text{A}{}_{\mathrm{?B}}\text{D}\text{→}\text{π}\text{Hom}{}_{\mathrm{A\prime }}\text{(B,D)}$

where

$\text{π(a ? d) = ab′d, a ∈ A, d ∈ D}\text{and}\text{B′ ∈ B′,}$

is injective.