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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.20363 |
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| _version_ | 1866909273657704448 |
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| author | Cortés-Izurdiaga, Manuel Poveda, Alejandro |
| author_facet | Cortés-Izurdiaga, Manuel Poveda, Alejandro |
| contents | Given a module $X$ and a regular cardinal $κ$ we study various notions of $(κ,\mathrm{Add}(X))$-freeness and $(κ,\mathrm{Add}(X))$-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial $κ^+$-generated, $(κ^+,\mathrm{Add}(X))$-free and $(κ^+,\mathrm{Add}(X))$-separable module. Our construction allows $κ$ to be singular thus extending \cite[Theorem~4.7]{CortesGuilTorrecillas}. Bearing on similar set-theoretic assumptions, we characterize when every module $X$ has a perfect decomposition. As a subproduct we show that Enoch's conjecture for classes $\mathrm{Add}(X)$ is consistent with ZFC -- a fact first proved by Šaroch \cite{Saroch}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost free modules, perfect decomposition and Enochs's conjecture Cortés-Izurdiaga, Manuel Poveda, Alejandro Rings and Algebras Logic Given a module $X$ and a regular cardinal $κ$ we study various notions of $(κ,\mathrm{Add}(X))$-freeness and $(κ,\mathrm{Add}(X))$-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial $κ^+$-generated, $(κ^+,\mathrm{Add}(X))$-free and $(κ^+,\mathrm{Add}(X))$-separable module. Our construction allows $κ$ to be singular thus extending \cite[Theorem~4.7]{CortesGuilTorrecillas}. Bearing on similar set-theoretic assumptions, we characterize when every module $X$ has a perfect decomposition. As a subproduct we show that Enoch's conjecture for classes $\mathrm{Add}(X)$ is consistent with ZFC -- a fact first proved by Šaroch \cite{Saroch}. |
| title | Almost free modules, perfect decomposition and Enochs's conjecture |
| topic | Rings and Algebras Logic |
| url | https://arxiv.org/abs/2407.20363 |