Is Zakat obligatory for the wife on her share of an apartment she inherited that is rented out, knowing that she does not work and the apartment's rent goes to her mother?

There is no zakat (definition) due on the wife for the rent she pays to her mother, nor is zakat obligatory on the rented apartment itself. Zakat becomes obligatory only on the rent if it reaches the nisab (definition) (minimum threshold) and a hawl (a full lunar year) has passed over it. Similarly, no zakat is due on the inherited house in which the mother-in-law resides.