Strongly clean matrices over arbitrary rings

We characterize when the companion matrix of a monic polynomial over an arbitrary ring R is strongly clean, in terms of a type of ideal-theoretic factorization (which we call an iSRC factorization) in the polynomial ring R[t]. This provides a nontrivial necessary condition for Mn(R) to be strongly clean, for R arbitrary. If the ring in question is either local or commutative, then we can say more (generalizing and extending most of what is currently known about this problem). If R is local, our iSRC factorization is equivalent to an actual polynomial factorization, generalizing results in [1], [18] and [12]. If, instead, R is commutative and h∈R[t] is monic, we again show that an iSRC factorization yields a polynomial factorization, and we prove that h has such a factorization if and only if its companion matrix is strongly clean, if and only if every algebraic element (in every R-algebra) which satisfies h is strongly clean. This generalizes the work done in [1] on commutative local rings and provides a characterization of strong cleanness in Mn(R) for any commutative ring R.