The Complexity of Module Radicals

Abstract

We construct a computable module $\mathcal{M}$ over a computable commutative ring R such that the radical of $\mathcal{M}$, $\mathrm{rad}(\mathcal{M})$, defined as the intersection of all proper maximal submodules, is ${\mathrm{\Pi }}_1^1$-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.