On ideals of product of commutative rings and their applications

TL;DR

This paper investigates the structure of ideals in product rings of commutative rings, establishing bounds on maximal ideals, topological properties, and applications to ring decompositions and continuous functions.

Contribution

It provides new bounds on the number of maximal ideals in product rings, explores topological homeomorphisms, and connects ring properties with continuous function spaces.

Findings

Number of maximal ideals in infinite product rings is at least 2^{2^{|A|}}.

Maximal ideals of component rings embed as closed subsets in the product ring.

Conditions under which rings are or are not direct summands of their ideals.

Abstract

In this paper, leveraging the recent achievements of researchers, we have revisited the family of ideals of product of commutative rings. We demonstrate that if $ \{ R_\alpha \}_{\alpha \in A} $ is an infinite family of rings, then $ \left| Max \left( \prod_{\alpha \in A} R_\alpha \right) \right| \geqslant 2^{2^{|A|}} $. Notably, if these rings are local then the equality holds. We establish that $ Max(R_\alpha) $ is homeomorphic to a closed subset of $ Max \left( \prod_{\alpha \in A} R_\alpha \right) $, for each $ \alpha \in A $. Additionally, we show that $ Max(R) $ is disconnected \ff $ R $ is direct summand of its two proper ideals. We deduce that if the intersection of each infinite family of maximal ideals of a ring is zero, then the ring is not direct summand of its two proper ideals. Furthermore, we prove that for each ring $R$, $ C\left(Max(R)\right) $ is isomorphic to $ C\left(Max\left(C(Y)\right)\right) $, for some compact $T_4$ space $Y$. Finally, we explore that $h_M(x)$'s can define roles of zero-sets.