The computational complexity of module socles
This paper studies the computational complexity of socles of modules from the viewpoint of computability theory. The socle Soc(M) of a module M over a commutative ring R is the sum of all simple submodules of M. Soc(M) plays a vital role in module theory and it is known as the largest semisimple sub...
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| Published in: | Annals of pure and applied logic Vol. 173; no. 5; p. 103089 |
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| Format: | Journal Article |
| Language: | English |
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Elsevier B.V
01.05.2022
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| ISSN: | 0168-0072 |
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| Abstract | This paper studies the computational complexity of socles of modules from the viewpoint of computability theory. The socle Soc(M) of a module M over a commutative ring R is the sum of all simple submodules of M. Soc(M) plays a vital role in module theory and it is known as the largest semisimple submodule of M. Socles of Z-modules (i.e., abelian groups) are Σ10 and there is a computable abelian group G such that Soc(G) is Σ10-complete. Socles of R-modules are Σ30. We show that there is a computable module M over a computable commutative ring R such that Soc(M) is Σ30-complete. |
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| AbstractList | This paper studies the computational complexity of socles of modules from the viewpoint of computability theory. The socle Soc(M) of a module M over a commutative ring R is the sum of all simple submodules of M. Soc(M) plays a vital role in module theory and it is known as the largest semisimple submodule of M. Socles of Z-modules (i.e., abelian groups) are Σ10 and there is a computable abelian group G such that Soc(G) is Σ10-complete. Socles of R-modules are Σ30. We show that there is a computable module M over a computable commutative ring R such that Soc(M) is Σ30-complete. |
| ArticleNumber | 103089 |
| Author | Wu, Huishan |
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| Cites_doi | 10.1215/00294527-2019-0036 10.1017/S0004972700031014 10.1090/proc/12509 10.1016/0003-4843(79)90011-1 10.1215/00294527-2021-0017 10.1007/978-3-642-31933-4 10.1016/j.jalgebra.2007.02.058 10.1016/j.jalgebra.2009.07.039 10.1016/0003-4843(77)90015-8 10.1017/bsl.2014.32 |
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| Keywords | 68Q15 Computability theory 03D80 68Q17 Socles of modules Simple module 03D15 |
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| Title | The computational complexity of module socles |
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