Strong cleanness of matrix rings over commutative rings

Communications in Algebra (Impact Factor: 0.39). 05/2008; 36(2). DOI: 10.1080/00927870701715175
Source: arXiv

ABSTRACT Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $\mathbb{M}_n(R)$ is strongly clean for each integer $n>0$ if $R$ is Henselian and we show that the converse holds if either the residue class field of $R$ is algebraically closed or $R$ is an integrally closed domain or $R$ is a valuation ring. It is also shown that each $R$-algebra which is locally a direct limit of module-finite algebras, is strongly clean if $R$ is a $\pi$-regular commutative ring.


Available from: François Couchot, Apr 04, 2015
1 Follower
  • [Show abstract] [Hide abstract]
    ABSTRACT: Let R be an associative ring with identity. An element a∈R is called clean if a=e+u with e an idempotent and u a unit of R and a is called strongly clean if, in addition, eu=ue. Aring R is called clean if every element of R is clean and R is strongly clean if every element of R is strongly clean. In the paper [Nicholson and Zhou, Clean rings: a survey, Advances in Ring Theory, 181–198, World Sci. Pub., Hackensack, NJ, 2005], the authors brought out an up to date account of the results in the study of clean rings. Here, we give an account of the results on strongly clean rings.
    Acta Applicandae Mathematicae 01/2009; 108(1):157-173. DOI:10.1007/s10440-008-9364-6 · 0.70 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: 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]R[t]. This provides a nontrivial necessary condition for Mn(R)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]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)Mn(R) for any commutative ring R.
    Journal of Algebra 02/2014; 399:854–869. DOI:10.1016/j.jalgebra.2013.08.044 · 0.60 Impact Factor