Ajax ( in Greek the name is Aias) , the name as the legend says comes from Heracles ,
When Aias was born Heracles was
feasting with his friend Telamon, king of Salamis, and he prayed to Zeus that the new born child be a strong and brave son , as he had spoken an eagle appeared so he named the child Aias after the bird.
Aias was one of Helens suitors , so he was bound by the Oath of Tyndareus to protect her . So when the time came to sail against Troy, Aias became Leader of the Salaminians, and contributed with 18 ships to the Achaean fleet .
During the Trojan War, Aias fought in single combat against Hector until nightfall so they parted them . The warriors, then, exchanged gifts: Hector gave him his sword, and Aias gave Hector his belt.
"Ever since I took into my hand this gift from Hector, my greatest enemy, it has brought me no good . Yes, men''s proverb is true: the gifts of enemies are no gifts and bring no good."
When Achilles was killed his body was dragged around the city walls by the Trojans as an insult to the dead. Aias seeing this risked his life to rescue the body and to honor the Greek army''s greatest General .
Achilles mother Thetida wanted to give her sons armor and weapons to the most worthy Greek warrior . But only two heroes were worthy enough to accept those gifts. One of them was Odysseus (Ullysses), the smartest Greek warrior, who had saved the Greek army many times from catastrophe in rough situations and Aias who was the strongest and bravest warrior after Achilles.
They were both very worthy of those gifts and neither Odysseus nor Aias wanted to admit the other’s superiority. They even took their swords out to compete for them but the others held them back. Then Nestor suggested they would let their enemies decide who deserved the gifts . So they sent two spies just outside the city walls of Troy to overhear the conversations of the Trojan people.
They heard two women talking about how Aias and Odysseus rescued Achilles body from the Trojan warriors and took it back to the Greek camp. One woman thought that Aias was stronger because he, under a shower of darts, carried the body, and fought at the same time. The other woman, with the help of Athena, said that anybody could do that, but Odysseus was the one that protected Aias while he went backwards and he fought all the other warriors. That made the other woman agree with her friend. The spies, after having heard all that, went back and told Nestor what happened.
So the Greek Kings decided that the armor and weapons should go to Odysseus.
Athena wanted Odysseus to get the armor and weapons for two reasons. She liked Odysseus more than Aias and she helped him all his life. The second reason is that Aias never asked for help from the Gods , once while he was in combat , Athena wanted to help him and give him courage , Aias told her to go away.
Aias was devastated by this, his pride and honor were now in crisis. He thought that everything he had done up to now would be forgotten and that nobody would know him now. His reaction was terrible, the Greek doctor Podalirios was the first one to notice the crisis he was going through . Aias wanted to kill all the Greek kings with his own hand during the night that was yet to come.
After night fell upon the camp and everybody was asleep, Aias with his sword in hand started to kill all the kings, as he had planned. Athena, seeing what was about to be done, blurred his mind and made him think that a flock of sheep were the Greek Kings. He killed all the sheep, thinking they were the kings and just left one, whom he thought was Odysseus. He took that sheep and tortured it in his tent.
When daybreak came the blur in his mind faded away and he saw what he had done. He felt very foolish and thought that everybody would laugh at him. So he decided to end his lifeslew himself by letting himself fall upon his sword (the sword he had received from Hector).
When the others found him, Agamemnon thought that what Aias had done was betrayal and he ordered that Aias was not to be honored , but to be put in a box and in the earth without any ceremonies. That was very dishonorable for a warrior at that time.
That’s how one of the greatest, strongest and bravest hero''s of the Trojan war died.
In a famous passage of the Odyssey, Odysseus sees Aias in the Underworld and tries to appease him, but Aias turns away in silence.