On the number of ring topologies on countable rings (Q2831436)

Let \(R\) be an infinite countable not necessarily associative ring and \(X(R)\) be the partially ordered set of all Hausdorff ring topologies on it. In the present paper it is proved that if \(\mathcal T\in X(R)\) is a metrizable and non-discrete topology, then: (i) there exists a chain of cardinality \(\mathfrak c=2^{\aleph_0}\) in \(X(R)\) such that each of its elements contains \(\mathcal T\) and (ii) there exists a subset \(Y\) of cardinality \(2^{\mathfrak c}\) in \(X(R)\) such that the supremum \(\mathcal T_1\vee\mathcal T_2\) is the discrete topology for any different topologies \(\mathcal T_1,\mathcal T_2\in Y\).