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## 2006.114: Ideals in mod-R and the ω-radical

2006.114: Mike Prest (2005) Ideals in mod-R and the ω-radical. Journal of the London Mathematical Society, 71 (2). pp. 321-334. ISSN 0024-6107

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## Abstract

Let $R$ be an artin algebra, and let mod-$R$ denote the category of finitely presented right $R$-modules. The radical ${\rm rad}={\rm rad}({\rm mod}\mbox{-}R)$ of this category and its finite powers play a major role in the representation theory of $R$. The intersection of these finite powers is denoted ${\rm rad}^\omega$, and the nilpotence of this ideal has been investigated, in $[{\bf 6}$, ${\bf 13}]$ for instance. In $[{/bf 17}]$, arbitrary transfinite powers, ${\rm rad}^\alpha$, of rad were defined and linked to the extent to which morphisms in ${\rm mod}\mbox{-}R$ may be factorised. In particular, it has been shown that if $R$ is an artin algebra, then the transfinite radical, ${\rm rad}^\infty$, the intersection of all ordinal powers of rad, is non-zero if and only if there is a ‘factorisable system’ of morphisms in rad and, in that case, the Krull–Gabriel dimension of ${\rm mod}\mbox{-}R$ equals $\infty$ (that is, is undefined). More precise results on the index of nilpotence of rad for artin algebras were proved in $[{\bf 14}$, ${/bf 20}$, ${/bf 24}\hbox{--}{/bf 26}]$.

Item Type: Article MSC 2000 > 16 Associative rings and algebras 2006.114 Professor Mike Prest 22 May 2006