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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Bibliographic Details
Published in:Annals of pure and applied logic Vol. 173; no. 5; p. 103089
Main Author: Wu, Huishan
Format: Journal Article
Language:English
Published: Elsevier B.V 01.05.2022
Subjects:
Computability theory
Simple module
Socles of modules
68Q15
Computability theory
03D80
68Q17
Socles of modules
Simple module
03D15
ISSN:0168-0072
Online Access:Get full text
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Summary: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.
ISSN:0168-0072
DOI:10.1016/j.apal.2022.103089