## 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

### Citations:

Zbl 1162.16016; Zbl 1162.16017; Zbl 1168.16014
Full Text:

### References:

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