We hear tons about Alexander's holdings in Egypt after his death and the Ptolemaic dynasty, but how did the other territory holdings fare? How long did they fall under Greek control or at least maintain ties with the Greek homeland? Which lasted the longest and how did they ultimately fail?
You should look up the Seleucid Empire, established by Alexander's general Seleucus. At its height it controlled much of the Near East, Persia and on into Central Asia. By 63 BC it was reduced to a rump state in Syria before finally being conquered and annexed by the Roman general Pompey.
Notable personalities include Seleucus, Antiochus III the Great and Antiochus IV Epiphanes (cf. the Maccabean Revolt).
I've found a few previous answers to get you started, by u/Daeres and u/Tigris_Vadam:
https://www.reddit.com/r/AskHistorians/comments/eeiodd/interested_in_the_seleucid_empire_any_book/