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Strongly clean matrices over arbitrary rings. (English) Zbl 1310.16023

Summary: 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 \(\mathbb M_n(R)\) 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 [G. Borooah et al., J. Pure Appl. Algebra 212, No. 1, 281-296 (2008; Zbl 1162.16016)], [X. Yang and Y. Zhou, J. Algebra 320, No. 6, 2280-2290 (2008; Zbl 1162.16017)] and [B. Li, Bull. Korean Math. Soc. 46, No. 1, 71-78 (2009; Zbl 1168.16014)]. If, instead, \(R\) is commutative and \(h\in R[t]\) 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 [G. Borooah et al., loc. cit.] on commutative local rings and provides a characterization of strong cleanness in \(\mathbb M_n(R)\) for any commutative ring \(R\).

MSC:

16S50 Endomorphism rings; matrix rings
16U60 Units, groups of units (associative rings and algebras)
13H99 Local rings and semilocal rings
15A18 Eigenvalues, singular values, and eigenvectors
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