"The professor's computer is expensive."
Translation:حاسوب اَلْأُسْتاذة غالي.
Why does حاسوب get no definite article here? Would الحاسوب الاستاذة be incorrect?
Yep, that would be incorrect.
Think of it that way: حاسوب and أستاذة are a compound (2 nouns added to each other to give a meaning). In such compounds, the definite article goes to the second noun in the compound. The addition of "AL" is pretty important here because it is the marker that tells that the first noun is added to the second noun (i.e. "AL" is the chain the links the two nouns). Hence: حاسوب الأستاذة (computer of the she-professor).