Ajax the Great was the King of Salamis and one of the primary participating commanders of the Trojan War. Of the Achaeans, he was considered lesser than only Achilles and Diomedes (who had bested him in a sparring duel). Ajax was a commanding figure on the battlefield and he received no notable injuries in the events described in the Iliad. He was also the only participating major figure on either side that did not received divine aid of some sort. Many later Athenian heroes traced their lineage back to Ajax the Great.
Ajax died by his own hand after losing a contest for Achilles' armor. He competed against Odysseus, and in the end, was "defeated only by his own sorrow". That event is described in Homer's Metamorphoses.
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Ajax won't approach Odysseus because he killed himself and died. There is no way for him to speak because he is already dead.
Ajax is a Greek hero found in both of Homer's stories, the Iliad and the Odyssey. In the Iliad, which is where Ajax met his fame, he is a highly regarded warrior. He was the King of Salamis and he led his solders to fight for Agamemnon in the war against Troy. He was an outstanding warrior, second only to the mighty Achilles in skill. Furthermore his is one of the only heroes in the Iliad who was unaided by any God or Goddess. Ajax survived the Trojan War. Shortly after the war there was a dispute between him and Odysseus about who should receive the shield and armor of Achilles. Odysseus received the honor, and in despair Ajax killed himself. In the Odyssey Ajax appears when Odysseus goes into Hades, where he finds Ajax mourning his own suicide.