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Free algebras, universal models and Bass modules https://arxiv.org/abs/2407.15864 #mathRA

Free algebras, universal models and Bass modules

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 $R^{(κ)}$ 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: $R^{(κ)}$ 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.

arxiv.org
July 25, 2024 at 3:10 AM · · feed2toot · 0 · 0 · 0
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