I am new to learning Latin. I have been doing exercises related to the genitive and got one wrong and I am not understanding why. The question was to translate the following into Latin - "We greet the master of the house." I answered "Villam domini salutamus." In the answer key they have "Dominum villae salutamus". Which is correct? Am I wrong? Doesn't the possessor go into the genitive? The house doesn't own the master...the master owns the house. Where am I confused? Thank you in advance for the help.
Possessive is different from "owning." The master owns the house (presumably), but the house has a master. It possesses a master, but it doesn't "own" it. Ownership is a legal thing, whereas the genitive case describes a grammatical relationship, and you'll see it in places where ownership makes no sense, like "for the love of money" ("of money" here is in the genitive) or "the color of the hair" ("of the hair" is the genitive), but money doesn't "own" the love, and the hair doesn't "own" the color.
Even in English, this is clear. The other way of writing "the master of the house" is "the house's master." House, not master, turns into the genitive case in English.
There are key words when translating from English. While you'll see plenty of exceptions later, if you see phrases like of the or the possessive 's (or s'), those generally are equivalent to the genitive case in Latin, regardless of who actually owns what.
As to your particular translation, villam domini salutamus means "we greet the house of the master." The direct object is what you put into the accusative case. You greet whom? You greet the master. Therefore regardless of "house," the basic sentence is dominum salutamus. Then you describe the master. The master of what? The master of the house, so dominum villae salutamus.