"The life of man [is] solitary, poor, nasty, brutish, and short."
And so here we go, Ajax and I, down the rabbit hole.
It is to be utterly expected that he lived the life of the thief, vandal and pirate: everybody did. So he learned to respect Agamemnon only because the High King was more powerful; every man has a master.
Other men like himself? They honored each others' exploits, their courage and success in raiding and battle, their reliability under fire. Brothers in arms.
Women? He competed for Helen because her property was better and easier to hold than his island. Hesione's fate is simple and convenient, and no reason to trouble himself about it. She is just another of the many creatures he has slaughtered for good reason: sacrifice, dinner. And his wife? Why and how could he ever have learned to be gentle about sex, when rape has always been and still is one of the most common tools of conquest? He's not conquering her; he already owns her. But what else does he know? After their experiences with the whores in Vietnam, it took US vets a long time to relearn gentle sex.
So what is his downfall? Did he seek some higher nobility? If so, where would he even have gotten that idea? What disinterestedly honorable dealing has he ever been involved in?
Perhaps only Hector, who he fought to a standstill and then they exchanged gifts, which must have startled him. Did he see Hector as an unappreciated and doomed workbeast? An ox or horse, driven, then beaten, to perform, but never honored? Did the one-time-only sight of Hector walking forth alone to meet Achilles with no one but his wife even watching, make Ajax's heart, like the Grinch's, grow three sizes that day?