"He does not like fish."

Translation: Він не любить рибу.

[lawnmoen]

Shouldn't fish decline to риби because the negation would mean it's in the genitive case?

[SergioRuido]

In this case not.

[lawnmoen]

Ok, could you explain why?

[helen.hein]

The negation isn't going to the fish, it's going to the liking (of the fish, in this case). He's not saying he doesn't have any fish (or whatever other thing you want to use); there are fish, and he just doesn't like them.
Does that make sense?

[SergioRuido]

This is advanced grammar. If you are interested - Google it. My priority for now is to process the reports and help learners who start from zero. Sorry.

[Vinnfred]

I think I should disagree...:)
http://kulturamovy.univ.kiev.ua/KM/pdfs/Magazine19-16.pdf

[Yoog]

I think that in case he doesn't like the fish (for example that one in the aquarium), we would say 'Він не любить рибу", but if fish means food (I guess that's the case in the sentence) it's rather 'Він не любить риби".

[spicy_wolf]

you are right. This course seems to negate some rules.

[helen.hein]

Realizing that this is for beginners, how important is word order?

[Vinnfred]

Because of the cases you can change the order and it will still mean the same thing. However, this order is the most natural one. In this sentence there are really no words that you can re-order and still get the natural sentence. But if you are writing some literature, especially poetry you can play with words and get different emphasis