Ajax is neither a god nor did he fight Greece..
He was a Greek himself, and he was a Greek hero (and cousin of Achilles) he was the strongest warrior greece had after Achilles died (Ajax retrieved Achilles' dead body after he was killed). He later commited suicide after he had murdered some sheep in a blind rage because he didn't get Achilles' armor (Odysseus got it even though Ajax had saved Achilles' body)
A little too lazy to log in to mention this (might be too late, too) But although Ajax was not a god he achieved something no other Greek or Trojan soldier could: he wounded a god. Ajax faced the rampaging god of war, Aries on the battlefield and struck him head on, wounding Aries so severely that the god fled from the field of war.
He was a Greek himself, and he was a Greek hero (and cousin of Achilles) he was the strongest warrior greece had after Achilles died (Ajax retrieved Achilles' dead body after he was killed). He later commited suicide after he had murdered some sheep in a blind rage because he didn't get Achilles' armor (Odysseus got it even though Ajax had saved Achilles' body)