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SUMMARY:Philipp Rothmaler (CUNY)
DTSTART:20220510T160000Z
DTEND:20220510T170000Z
DTSTAMP:20260423T021403Z
UID:ONCAS/18
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ONCAS/18/">E
 very module has an Ulm length</a>\nby Philipp Rothmaler (CUNY) as part of 
 ONCAS Online Noncommutative Algebra Seminar\n\n\nAbstract\nTo make sense o
 f the statement in the title\, I introduce a concept of Ulm submodule that
  grew out of discussions with Alex Martsinkovsky and applies to any module
  over any associative ring with 1. In abelian groups it coincides with the
  classical notion\, and so does the rest of the investigation. \n\nAs usua
 l\, a module is said to have Ulm length 0 if it coincides with its own Ulm
  submodule. These modules form a definable subcategory\, which\, over doma
 ins\, coincides with that of divisible modules. The subcategory is equal t
 o the entire category if and only if the ring is absolutely pure (on the s
 ame side). Over RD-domains\, like Prüfer domains or the first Weyl algebr
 a over a field of characteristic 0\, Ulm length 0 modules are injective.\n
 \nTaking the Ulm submodule constitutes a functor\, which\, by iteration\, 
 leads to higher Ulm functors as usual and in turn to Ulm sequences and the
  notion of Ulm length for any module. One of the main results is that the 
 (first) Ulm submodule of a pure-injective has length 0. In other words\,  
 pure-injectives have Ulm length at most 1\, just as over the integers. As 
 a consequence\, every module is a pure (even elementary) submodule of a mo
 dule of Ulm length at most 1.\n\nAs another consequence one obtains\, for 
 any pure-injective over an RD domain\, a direct decomposition into a large
 st injective submodule (= the first Ulm submodule)\, and a reduced module\
 , that is\, a module with zero Ulm submodule.\n
LOCATION:https://researchseminars.org/talk/ONCAS/18/
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