Linear maps preserving matrices annihilated by a fixed polynomial

Let M n ( F ) be the algebra of n × n matrices over an arbitrary field F . We consider linear maps Φ : M n ( F ) → M r ( F ) preserving matrices annihilated by a fixed polynomial f ( x ) = ( x − a 1 ) ⋯ ( x − a m ) with m ≥ 2 distinct zeroes a 1 , a 2 , … , a m ∈ F ; namely, f ( Φ ( A ) ) = 0 whenever f ( A ) = 0 . Suppose that f ( 0 ) = 0 , and the zero set Z ( f ) = { a 1 , … , a m } is not an additive group. Then Φ assumes the form (†) A ↦ S ( A ⊗ D 1 A t ⊗ D 2 0 s ) S − 1 , for some invertible matrix S ∈ M r ( F ) , invertible diagonal matrices D 1 ∈ M p ( F ) and D 2 ∈ M q ( F ) , where s = r − n p − n q ≥ 0 . The diagonal entries λ in D 1 and D 2 , as well as 0 in the zero matrix 0 s , are zero multipliers of f ( x ) in the sense that λ Z ( f ) ⊆ Z ( f ) . In general, assume that Z ( f ) − a 1 is not an additive group. If Φ ( I n ) commutes with Φ ( A ) for all A ∈ M n ( F ) , or if f ( x ) has a unique zero multiplier λ = 1 , then Φ assumes the form (†) . The above assertions follow from the special case when f ( x ) = x ( x − 1 ) = x 2 − x , for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map Φ : M n ( F ) → M r ( F ) sending disjoint rank one idempotents to disjoint idempotents always assume the above form (†) with D 1 = I p and D 2 = I q , unless M n ( F ) = M 2 ( Z 2 ) .

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