There are two Ajaxes in the Iliad. One is Ajax the Greater and the other is Ajax the Lesser. The Greater is the son of Telamon and the Lesser is the son of Oileus. Both come to bad ends in the Trojan War legends that form sequels to the Iliad. In the Iliad, they often fight together and it is easy to miss which Ajax “the bard known as Homer” is referring to, yet when people say “Ajax,” they invariably mean Ajax the Greater of Salamis.
Ajax the Lesser was from Locria. He brought 40 ships to Troy and was almost as fast as Achilles. Despite his sizable contribution to the Achaean efforts in the Trojan War, the Locrian Ajax is known principally for something he may not have done: the rape of Cassandra. Cassandra, the daughter of Priam and Hecuba who could reveal the future but would not be believed, had fled for safety to the temple of Athena at the end of the war. It appears that it was generally accepted that Ajax pulled her from the temple, but whether or not he actually raped her at the temple was disputed. He was acquitted, but Athena made sure he died anyway, impaled on what became known as the Rocks of Ajax.
Ajax the Greater, king of Salamis, was a large man, with a large thick shield, tremendous courage and stamina, and second to Achilles in battle prowess. When Achilles died Ajax and Odysseus recovered the body and both claimed the right to the hero’s armor. When Odysseus won it, Ajax went insane. Literally. He tries to kill the Achaean leaders, but only manages to kill some sheep because Athena, protecting her favorites, has deluded poor Ajax. When he regains his senses, he commits suicide.