The definite/indefinite article -- the/a(n) -- always comes before a noun and can never be used without a noun.

Is the definite/indefinite article a complement or a modifier of a noun?

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    @John Lawler, in a comment at How to determine if a-pre-head dependent of a noun is a complement or a modifier, rejects both the terminology and any treatment by a grammar that does not provide syntactic tests to distinguish their brand of 'complements' and 'modifiers'. He also says that a binary system is too broad-brush here. – Edwin Ashworth Apr 18 '18 at 10:28
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    @JK2 Ignore the useless negative comments. CGEL supports all its claims with explanations and evidence. I've just read (again) the section on attributive modifiers vs complements of nouns and they make the distinction abundantly clear. – BillJ Apr 18 '18 at 12:36
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    No, Complement and Modifier are two types of syntactic function (or grammatical relations). The term Determiner, like the word Subject refers to a different type of syntactic function not covered by the other two terms. If one accepts that noun phrases are phrases ultimatley headed by nouns, then what Complements, Modifiers and Determiners have in common is that they are all dependents (i.e. not Heads). – Araucaria Apr 18 '18 at 13:24
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    @Araucaria So you're saying that determiners (and of course including articles) are neither complements nor modifiers of nouns? – JK2 Apr 18 '18 at 15:36
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    @JK2 Yes, exactly so (according to writers such as H&P). – Araucaria Apr 18 '18 at 15:40

A modifier changes the meaning of an element. https://en.wikipedia.org/wiki/Grammatical_modifier
An article is a determiner. It serves to express a noun's or a noun phrase's reference in the context. https://en.wikipedia.org/wiki/English_determiners

  • 'Red' in 'a red book' is a modifier, but it adds further detail (and can be used to restrict a set [here, of all books present] – when used in the identifier role) rather than changing the meaning of 'book'. 'Fake' in 'a fake diamond' certainly changes the meaning. – Edwin Ashworth Apr 18 '18 at 10:48
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    "Fake" describes the quality of the diamond. And "red" also describe the quality of the book, "the book is red". But "a" in "a book" tells us nothing about the object's qualities or properties, it only tells us that the object is "any book you can imagine". And "the" in "the book" also adds nothing to the description of the object itself. It tells us something like "book that has been mentioned somewhere above". – Roman Ryzhiy Apr 18 '18 at 11:05
  • @RomanRyzhiy Your missing Ed's point there. A fake diamond is NOT a diamond, whereas a red diamond IS a diamond! – Araucaria Apr 18 '18 at 13:21
  • What I'm saying is that a modifier (linguistics sense) (here, I'll confine the argument to 'modifier of noun') doesn't need to modify (default everyday sense: make minor changes to) the noun it attaches to. And if there are changes, they can be drastic (as with the [fake] diamond). – Edwin Ashworth Apr 18 '18 at 13:45

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