Every module has an Ulm length

Abstract: To make sense of 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.

As usual, 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 domains, coincides with that of divisible modules. The subcategory is equal to the entire category if and only if the ring is absolutely pure (on the same side). Over RD-domains, like Prüfer domains or the first Weyl algebra over a field of characteristic 0, Ulm length 0 modules are injective.

Taking 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 module of Ulm length at most 1.

As another consequence one obtains, for any pure-injective over an RD domain, a direct decomposition into a largest injective submodule (= the first Ulm submodule), and a reduced module, that is, a module with zero Ulm submodule.

rings and algebrasrepresentation theory

Audience: researchers in the topic

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