Two classical splitting theorems: Easy proofs

AbstractThe 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 A⊗BD→πHomA′(B,D) where π(a ⊗ d) = ab′d, a ∈ A, d ∈ D and B′ ∈ B′, is injective