"The mother of Carmen has a new telephone."
Translation:Mama lui Carmen are un telefon nou.
Yes. Female names not ending in a vowel and names that came from another language use the possessive pronoun "lui". For example: lui Carmen (Carmen's), lui Ingrid, lui Naomi, lui Beatrice etc.
here are some "normal" names:
Andreea - Andreei
Roxana - Roxanei
Iulia - Iuliei
Ioana - Ioanei
Maria - Mariei
Dana - Danei
Alexandra - Alexandrei etc.
Cu plăcere! The thing is you have to learn them "as is". I don't know how to better put it but as a native "Carmenei" sounds awful. Same as "Beatricei" or "Ingridei". The same is true if you use "Lui Maria" "Lui Andreea", it sounds wrong and it's considered a mistake.
As a rule of thumb, all native romanian female names end in the vowel "a". So if a name ends in "a" you pretty much know you don't use "lui".