We define various ring sequential convergences on $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence ${𝕃}_1$ on $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields $\mathbb{Z}/\left(p\right)$. Further, we show that $(\mathbb{Z},{𝕃}_1^{*})$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.

In this paper, it is proved that a first-countable paratopological group has a regular $G_{\delta }$-diagonal, which gives an affirmative answer to Arhangel’skii and Burke’s question [Spaces with a regular $G_{\delta }$-diagonal, Topology Appl. 153 (2006), 1917–1929]. If $G$ is a symmetrizable paratopological group, then $G$ is a developable space. We also discuss copies of $S_{\omega }$ and of $S_2$ in paratopological groups and generalize some Nyikos [Metrizability and the Fréchet-Urysohn property in topological groups, Proc. Amer. Math....