Characterizing categoricity in the class $Add(M)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917361887477760 |
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| author | Zhang, Xiaolei |
| author_facet | Zhang, Xiaolei |
| contents | We show that the condition of being categorical in a tail of cardinals can be characterized for the class of $R$-modules of the form $\Add(M)$. More precisely, let $R$ be a ring and $M$ be an $R$-module which can be generated by $\leq \aleph$ elements. Then $\Add(M)$ is $κ$-categorical in all $κ>\Vert R\Vert+\aleph+\aleph_0$ if and only if $\Add(M)$ is $κ$-categorical in some $κ>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $κ$ in $\Add(M)$ is $M$-free for all $κ>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $(\Vert R\Vert+\aleph+\aleph_0)^{+}$ in $\Add(M)$ is $M$-free.
As an application, we show that the class of pure-projective $R$-modules is categorical in some (all) big cardinal if and only if the module $P^{(\aleph_0)}$ is free for each countably generated pure-projective $R$-module $P$; the class of semisimple $R$-modules is categorical in some (all) big cardinal if and only if $R$ admits a unique simple module up to isomorphism, partly answering a question proposed in [5, Mazari-Armida M., Characterizing categoricity in several classes of modules. J. Algebra 617, 382-401 (2023)]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_19641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing categoricity in the class $Add(M)$ Zhang, Xiaolei Rings and Algebras We show that the condition of being categorical in a tail of cardinals can be characterized for the class of $R$-modules of the form $\Add(M)$. More precisely, let $R$ be a ring and $M$ be an $R$-module which can be generated by $\leq \aleph$ elements. Then $\Add(M)$ is $κ$-categorical in all $κ>\Vert R\Vert+\aleph+\aleph_0$ if and only if $\Add(M)$ is $κ$-categorical in some $κ>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $κ$ in $\Add(M)$ is $M$-free for all $κ>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $(\Vert R\Vert+\aleph+\aleph_0)^{+}$ in $\Add(M)$ is $M$-free. As an application, we show that the class of pure-projective $R$-modules is categorical in some (all) big cardinal if and only if the module $P^{(\aleph_0)}$ is free for each countably generated pure-projective $R$-module $P$; the class of semisimple $R$-modules is categorical in some (all) big cardinal if and only if $R$ admits a unique simple module up to isomorphism, partly answering a question proposed in [5, Mazari-Armida M., Characterizing categoricity in several classes of modules. J. Algebra 617, 382-401 (2023)]. |
| title | Characterizing categoricity in the class $Add(M)$ |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2502.19641 |