2025/06/10 by Badie, Mehdi, Aliabad, Ali Rezaie, Obeidavi, Foad
#FOS: Mathematics #General Topology (math.GN) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2506.08537
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 \ Rα\α∈ A is an infinite family of rings, then | Max ( ∏α∈ A Rα) | \geqslant 2^2|A| . Notably, if these rings are local then the equality holds. We establish that Max(Rα) is homeomorphic to a closed subset of Max ( ∏α∈ A Rα) , for each α∈ 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(Max(R)) is isomorphic to C(Max(C(Y))) , for some compact T4 space Y. Finally, we explore that hM(x)'s can define roles of zero-sets.