"This is Professor Rania's room."
Translation:هٰذِهِ غُرْفة اَلْأُسْتاذة رانْيا.
OK, really confused now. With 'This' (هٰذِهِ) I thought that it was followed by 'a..' unless the following word had 'The' (اَل) in front of it. But when showing possession of the noun, can just be followed by the possessor (which doesn't have 'The', unless really meaning to say 'The') So shouldn't this be > هٰذِهِ اَلْغُرْفة أُسْتاذة رانْيا this is the room of professor rania / this is professor rania's room But I'm reading the above as; This is a room of (belonging to) the professor rania
Well, I'm kind of confused here as well by the question but I'll try to elaborate.
What we are about here is a Genitive relation between 2 nouns: غرفة (room) and أستاذة (professor/teacher ..etc).
In ANY Genitive relation (or as some like to call it: OF-relation) between two nouns in Arabic, the definite Article (if exists) goes on SECOND term of the compound. Thus, The room of the teacher is غرفة الأستاذة (ğurfatul-ustáTHah). It is not like the case with English where THE comes before both nouns. In Arabic, (AL الـ) only comes to the second term (i.e. in English, the noun that comes after OF).
Now, adding هذه (This, feminine) does not change the relation between the two nouns. This is just an additive and it is feminine because it refers to غرفة (room) which is a feminine noun in Arabic.
Lastly, the name Rania is just a name explaining who is the teacher. So, it comes after الأستاذة (meaning as if الأستاذة is just a title for Rania anyway). So, the main bond here or the core of the sentence is the Genitive or OF-relation between غرفة and أستاذة which when merged become غرفة الأستاذة - the room of the teacher/ the teacher's room (or just change teacher with professor as you like).
Let's remove (Rania) from the equation for a moment.
As per the genitive rule:
- A teacher's room: غرفة أستاذة.
- THE teacher's room: غرفة الأستاذة.
Now, since we are talking about a specific person (Rania) and "teacher" أستاذة acts as a title for her, then logically it should be defined: غرفة الأستاذة رانيا
This said, in dialects now this rule does really hold and people are easy about it. In dialects, you can indeed just say غرفة أستاذة رانيا but this is not exactly how it should be done in MSA (because in this manner it sounds as if أستاذة is part of the name of رانيا and not a title, but dialects are easy with that).