The characteristic polynomial of an algebra and representations

We pose two new linear preserver problems. First, given a field k and a finite dimensional k-algebra B we show that the only linear maps ϕ:k×d→B which preserve the unit and the n-th roots of unity (for some n>2 coprime to char(k)) are the algebra homomorphisms. Second, we consider linear maps ϕ:A→B between finite dimensional k-algebras which preserve the Cayley-Hamilton relations. We show that if preservation of the Cayley-Hamilton property is understood in a certain non-commutative sense, and A=k×d, then the only such linear mappings are the algebra homomorphisms.