The computational complexity of module socles - Wu, Huishan (Annals of pure and applied logic, 2022)
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.