Singular or plural noun form when relating [noun 1] and [noun 2]?

Hi!

I have some difficulties understanding the type of sentence contruction where one noun is followed by the word "of" and then another noun, for example in sentences like this:

The flame of the lamps flickered against the darkness.

Here "flame" can be said to be noun 1 and lamps noun 2.
So my question is here, how do you know whether noun 1 as well as noun 2 should take singular or plural form?
For example, the above sentence comes from a published novel, so I assume it's grammtical.

However, to me this sentence sounds quite strange, as if there is only one "flame" as a singular entity, rather than multiple flames as in one flame per lamp (and there are multiple lamps). Why isn't the sentence written as "The flames of lamps ..." instead?

I also have difficulty understanding other, similar sentences relating two nouns, such as:

Should it be:
The schoolchildren brought their textbook to the class?
or
The schoolchildren brought their textbooks to the class?

(assuming each schoolchild brought one book to the class, the same book as in authored by the same person, same edition, etc.)

Why?

Thanks a lot if someone could explain the grammar rule applicable to cases like this (could you please refer to some good grammar book section too), so I can understand how to correctly apply it to write sentences like the above and other similar ones relating nouns to one another!