In some cases, the article for singular countable nouns is ommited in Greek. For you to always be sure whether or not to add the article in english, check if the english translation makes sense without the article as well (in this case, it doesn't.) One of these cases is probably when the object of the sentence is supposed to be only one (based on logic), so the article is implied instead of added.
exp. "I'm eating an apple - Τρώω ένα μήλο/Τρώω μήλο",
"I'm writing a book - Γράφω ένα βιβλίο/Γράφω βιβλίο",
"I'm watching a film - Βλέπω μία ταινία/Βλέπω ταινία". I hope this helped a bit. :)
I understand "I'm not writing a book" is right, but shouldn't "I don't write books" be also accepted? Can't this sentence also convey this meaning? Because it seems to me that singular direct objects in greek can also mean general ideas that in english would be written in plural, like "Τρώω μήλο". Is that correct?
Perhaps "Τρώω μήλο" is not a good example because 'apple' and 'an apple' and 'apples' are all possible in English, where 'apple' is the stuff or the flavour. In other words, in 'I eat apple', 'apple' is the stuff, not the fruit, whereas in Greek, 'Τρώω μήλο' means "I am eating an apple'.
Indefinite article The indefinite article is identical to the number one and has only singular. The use of the indefinite article is dictated by the circumstances of the situation and contrary to English is only used if you want to specify the number.
"Αγόρασα υπολογιστή." ("I bought a computer") Since I only bought one there is no need to specify it. If you were to use the indefinite article it would show emphasis.
Often, there is no indefinite article in Greek where you would find one in English because it specifically expresses the concept of "one".
It often sounds odd to English speaker used to the indefinite article.
"Είναι δικηγόρος." "He is a lawyer" He can't be more than one. Τι καλό παιδί! "What a good boy!" We are speaking to/about one boy.
So, if you are not indicating a number you do not use the indefinite article. And if you do not have any article definite or indefinite you assume it is indefinite and it's only one item. In which case you would use "a/an" in English.