Exercise 2, § 14

Let K be a commutative field, A the K-algebra $\mathbf{M}_5$(K), B the subalgebra of matrices of the form

$x$ 0 0

0 Y 0,

0 0 Y

and C the subalgebra of matrices of the form

$x$ 0 0 0

0 $x$ 0 0

,

0 0 $x$ 0

0 0 0 Y

where $x$ runs through K and Y through the algebra $\mathbf{M}_2(K)$. Prove that B and C are semisimple algebras containing the center Z of A. Then, prove that there exists an isomorphism of K-algebras $f: B\rightarrow C$ that leaves invariant the elements of Z but that there does not exist any automorphism of A extending $f$ (observe that such an automorphism must send a simple component of B to a simple component of C).