We construct a computable module over a computable commutative ring R such that the radical of , , defined as the intersection of all proper maximal submodules, is -complete. This shows that in general such radicals are as (logically) complicated as possible and, unlike many other kinds of ring-theoretic radicals, admit no arithmetical definition.
"The Complexity of Module Radicals." Notre Dame J. Formal Logic 62 (2) 353 - 368, May 2021. https://doi.org/10.1215/00294527-2021-0017