Free algebras, universal models and Bass modules
| Autor: | Pillay, Anand, Rothmaler, Philipp |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $R^{(\kappa)}$ embeds every $\kappa$-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 $R^{(\kappa)}$ whose projectivity is equivalent to left perfectness, which allows to add a "stronger" equivalent condition: $R^{(\kappa)}$ embeds every $\kappa$-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. Comment: 19 pages |
| Databáze: | arXiv |
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