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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.15864 |
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| _version_ | 1866929661194272768 |
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| author | Pillay, Anand Rothmaler, Philipp |
| author_facet | Pillay, Anand Rothmaler, Philipp |
| contents | We investigate the question of when free structures of infinite rank (in a variety) possess model-theoretic properties like categoricity in higher power, saturation, or universality. Concentrating on left $R$-modules we show, among other things, that the free module of infinite rank $R^{(κ)}$ purely embeds every $κ$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory $T$ of $R^{(κ)}$ whose projectivity is equivalent to left perfectness, which allows to add a "stronger" equivalent condition: $R^{(κ)}$ purely (equivalently, elementarily) embeds every $κ$-generated flat left $R$-module which is a model of $T$.
In addition, we extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15864 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Free algebras, universal models and Bass modules Pillay, Anand Rothmaler, Philipp Rings and Algebras We investigate the question of when free structures of infinite rank (in a variety) possess model-theoretic properties like categoricity in higher power, saturation, or universality. Concentrating on left $R$-modules we show, among other things, that the free module of infinite rank $R^{(κ)}$ purely embeds every $κ$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory $T$ of $R^{(κ)}$ whose projectivity is equivalent to left perfectness, which allows to add a "stronger" equivalent condition: $R^{(κ)}$ purely (equivalently, elementarily) embeds every $κ$-generated flat left $R$-module which is a model of $T$. In addition, we extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules. |
| title | Free algebras, universal models and Bass modules |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2407.15864 |