What is the ruling on divorcing one's wife in writing over the internet, and what are the consequences in the two aforementioned cases: first, with the intention of jesting, and second, with the intention of divorce?

Writing a divorce online is like writing it [on paper]; it does not take effect unless the husband intends it. So, if you intended it, it takes effect.

If the divorce is revocable (one or two, and not preceded by a [final] divorce), you can reconcile with your wife before the 'iddah (waiting period) ends.

If the divorce is irrevocable (the third), she becomes unlawful for you and does not become lawful again unless she marries another man and he consummates the marriage with her, and then he divorces her.

The 'iddah ends with the wife's purification from the third menstruation, or by giving birth for a pregnant woman, or after three months for a woman who does not menstruate.