Nil-clean Companion Matrices

The classes of clean and nil-clean rings are closed with respect standard constructions as direct products and (triangular) matrix rings, cf. [12] resp. [4], while the classes of weakly (nil-)clean rings are not closed under these constructions. Moreover, while all matrix rings over fields are clean, [12] when we consider nil-clean rings there are strongly restrictions: if a matrix ring over a division ring F is nil-clean then F has to be isomorphic to F2, [11]. It can be useful to know the (nil-)clean elements in some rings which are not (nil-)clean. For instance, strongly clean matrices (i.e. they have a decomposition r = u + e such that eu = ue) over commutative local rings are studied in [7]. In particular, it would be nice to characterize nil-clean elements in matrix rings over division rings. For the case of strongly nil-clean elements (i.e. they have a decomposition r = e + n such that en = ne) we refer to [10, Theorem 4.4]. From this result we conclude that an n × n matrix over a division ring D is strongly nil-clean if and only if its characteristic polynomial has the form X(X − 1). For other studies of (nil-)clean elements in various rings we refer to [2] and [6]. Since in the proofs of the fact that the matrix ring Mn(F ) over the field F (resp. F = F2) is (nil-)clean the Frobenius (rational) normal form is used, it is useful to know when a companion matrix is nil-clean. In the main result of the present paper (Theorem 6) we characterize companion matrices over fields which are nil-clean. Moreover, it is proved that all these matrices have nil-clean like decompositions: for every polynomial χ of degree n such that the coefficient of X is 0, all nil-clean matrices can be decomposed as A = E + B where E is an idempotent and B is a matrix whose characteristic polynomial is χ. In fact the