The Complexity of Decomposability of Computable Rings

Abstract

This article studies the complexity of decomposability of rings from the perspective of computability. Based on the equivalence between the decomposition of rings and that of the identity of rings, we propose four kinds of rings, namely, weakly decomposable rings, decomposable rings, weakly block decomposable rings, and block decomposable rings. Let $\mathcal{R}$ be the index set of computable rings. We study the complexity of subclasses of computable rings, showing that the index set of computable weakly decomposable rings is m-complete ${\mathrm{\Sigma }}_1^0$ within $\mathcal{R}$ and that the index set of computable decomposable (resp., weakly block decomposable, block decomposable) rings is m-complete ${\mathrm{\Sigma }}_2^0$ within $\mathcal{R}$.