The traditional pronunciation for the detergent brand (and I think also the Greek hero, contrary to FX_'s answer) is
/ˈejdʒæks/ (often written
/ˈeɪdʒæks/), which can serve as guidance.
I believe the correct pronunciation is Ay-Jacks.
My answer is a tweaked version of the answer found here.
The traditional pronunciation for the Greek hero is
ˈeɪdʒaks, which can serve as guidance.
The pronunciation I find on the NOAD is /ˈeɪˌdʒæks/, which matches as a friend of mine (who is American) pronounces the word, independently from the meaning of the word.
aɪ.əs are right because AJAX is referred to the mythological Greek hero.
I prefer to say
aɪ.əs because is more similiar to the latin word
Aiax and to the italian word
"A" like in "way" and "asynchronous." "JAX" like "jacks" or "Jack's." This comes from my experience as a programmer and a computer science student. Since it's an acronym and it's not very common in speech, multiple pronunciations could be acceptable.
The Longman Pronunciation Dictionary suggests ['eɪdʒ æks] or ['aɪ æks].