A constructive counterpart of the subdirect representation theorem for reduced rings

TL;DR

This paper provides a constructive version of the subdirect representation theorem for reduced rings, enabling constructive proofs of related properties and extending to semiprime ideals.

Contribution

It offers a constructive counterpart to a classical theorem, facilitating constructive proofs of commutativity for rings with specific identities.

Findings

Constructive proof that every reduced ring is a subdirect product of domains.

Demonstration that rings satisfying x^3 = x are commutative.

Extension of results to semiprime ideals.

Abstract

We give a constructive counterpart of the theorem of Andrunakievi\v{c} and Rjabuhin, which states that every reduced ring is a subdirect product of domains. As an application, we extract a constructive proof of the fact that every ring $A$ satisfying $\forall x\in A. x^3=x$ is commutative from a classical proof. We also prove a similar result for semiprime ideals.