Breakdown of Wir haben kein Interesse an dem Idol.
haben
to have
kein
no, not any
wir
we
an
at, on (vertical) / to (addressee) / of (in quantity/stock expressions)
Questions & Answers about Wir haben kein Interesse an dem Idol.
Why does Interesse use the preposition an and why is it followed by the dative case here?
The noun das Interesse has a fixed partnership with the preposition an when expressing an interest in something. Because an is a two-way preposition and having an interest is a state rather than physical movement, it triggers the dative case, resulting in an dem Idol.
The verb sich interessieren uses für. Can I use für with the noun Interesse in this sentence?
No, you have to switch prepositions. While the verb sich interessieren pairs with für plus the accusative case (like in Wir interessieren uns für das Idol), the noun das Interesse takes an plus the dative case.
Could I combine an dem into am in this sentence?
Yes! The preposition an and the dative neuter article dem frequently contract into am. Saying Wir haben kein Interesse am Idol is perfectly correct and sounds very natural.
Why do we use kein here instead of nicht to say 'no interest'?
In German, kein is used to negate nouns that would normally appear with an indefinite article or no article at all. Since you are negating the noun Interesse rather than the verb haben, you must use kein.