This work is a continuation of studies that were in the work [1]. Definition. Let (Δ,<) be a lattice and let a, b ∈ Δ. If a < b and between elements a and b there exist no other elements in the lattice Δ then we shall say that the element b covers the element a in the lattice Δ. We shall denote this by a ≺Δ b. Remark 1. If I is an ideal of a ring R then the set {I} is a basis of filter of neighborhoods of zero for a ring topology. We shall denote this topology by τ (I).Theorem Let Q be a ring and let Δ be the lattice of all ring topologies on the ring Q or the lattice of all ring topologies of the ring Q in each of which the topological ring has a basis of filter of neighborhoods of zero which consists of subgroups of the additive group of the ring Q. If τ1 and τ2 are ring topologies such that τ1 ≺Δ τ2 and {Uγ|γ ∈ Γ} and {Vβ|β ∈ B} are bases of filters of neighborhoods of zero in topological rings (Q, τ1) and (Q, τ2), respectively, then for any ideal I of ring Q the following statements are true: 1. There exist ring topologies ¯τ1 and ¯τ2 such that families {I + Uγ|γ ∈ Γ} and {I +Vβ|β ∈ B} are bases of filters of neighborhoods of zero in the topological rings (Q, ¯τ1) and (Q, ¯τ2) respectively and either ¯τ1 ≺Δ ¯τ2 or ¯τ1 = ¯τ2. 2. Either sup{τ (I), τ1} = sup{τ (I), τ2} or the set {(Uγ  I) + Vβ|γ ∈ Γ,β ∈ B} will be a basis of filter of neighborhoods of zero in the topological ring (Q, τ1). 3 If I is an ideal of the ring Q such that sup{τ1, τ(I)} ̸= sup{τ2, τ(I)} and I is an open ideal in the topological ring (Q, τ2), then I is an open ideal in the topological ring (Q, τ1) too. Remark 2. A ring R, an ideal I and two ring topologies τ1 and τ2 in each of which the topological ring has a basis of filter of neighborhoods of zero which consists of subgroups of the additive group of the ring are constructed such that τ1 ≺Δ τ2 and between topologies sup{τ (I), τ1} and sup{τ (I), τ2} there are infinite number of ring topologies in each of which the topological ring has a basis of filter of neighborhoods of zero which consists of subgroups of the additive group of the ring. This example shows that the given in [1] conditions under which the properties of a unrefinable chain of ring topologies, are preserved under taking the supremum are essential.