The maximum ring topology on the rational number field among those for which the sequence \(1/n\) converges to zero. (Q1873301)

Let \((a_n)_{n\in\mathbb N}\) be a certain sequence of rationals. Write \(\tau (a_n)\) for the finest of all the ring topologies \(\tau\) on \(\mathbb Q\) such that \(a_n\to 0\) in \((\mathbb Q, \tau)\). In the article a basis of neighbourhoods of zero in \((\mathbb Q, \tau (\frac 1n))\) is constructed and it is proved that \(\tau (\frac 1n)\) is the finest of all the group topologies \(\mu\) on \(\mathbb Q\) such that \(\frac 1n\to 0\) in \((\mathbb Q, \mu)\). The article also contains a description of basis of neighbourhoods of zero for \((\mathbb Q, \tau (a_n))\) for certain sequences of rationals \((a_n)_{n\in\mathbb N}\), each of which is not the the finest of all the group topologies \(\tau\) on \(\mathbb Q\) such that \(a_n\to 0\) in \((\mathbb Q, \tau)\).