On ideals of product of commutative rings and their applications

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.