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


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.

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Available from: François Couchot, Apr 04, 2015
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    • "On montre que les anneaux pour lesquels les modules indécomposables vérifient une condition de finitude (de type fini, de type cofini, noethérien, artinien, de longueur finie, etc...) sont les anneaux de dimension de Krull nulle avec un radical de Jacobson T-nilpotent. En outre, dans [19], on montre qu'un anneau arithmétique R a ses anneaux de matrices fortement propres (tout élément est somme d'un idempotent et d'un élément inversible qui commutent) si et seulement si ses localisés en ses idéaux maximaux sont henséliens. -Dans [15] on étend aux anneaux commutatifs semihéréditaires les résultats sur les modules de présentation finie obtenus par Brewer, Katz, Klinger, Levy et Ullery sur les anneaux de Prüfer intègres ([3], [2] et [30]). "
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    ABSTRACT: This memoir is a presentation of the works carried out by the author in ring and module theory. More precisely the author studied arithmetical commutative rings, in particular valuation rings (which are not necessarily integral domain). The most remarkable result is the proof of the following fact: any local ring of bounded module type is an almost maximal valuation ring. Are also presented some results on the localization of injective modules and on the pure-injective hulls of some classes of modules.
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    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.
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