The predicative noun is used with the indefinite article if the speaker states that the object denoted by the noun belongs to a certain class.

If a predicative noun is modified by a particularizing attribute, the definite article is used.

He is the student you wanted to speak to.

If a predicative noun denotes a post that can be occupied by one person at a time, either no article or the definite article is used.

Mr. Henderson is manager, not under-manager any longer. (Lindsay)

Montanelli was director of the theological seminary at Pisa.  Comrade N. is the dean of our faculty.

No article is used with predicative nouns after the verbs to turn, to commence, to appoint, to elect.

Compeyson turned traitor.

Shakespeare commenced actor.

They appointed him head-teacher.

A predicative noun sometimes has an adjectival character, especially when it is followed by the adverb enough. In this case, no article is used.

Surely Bolla isn’t fool enough to believe that sort of stuff? (Voynich)

When a predicative noun in an adverbial clause of concession is placed at the head of the clause, no article is used.

Child as he was, David understood that Creakle was an ignorant man.

The nouns son and daughter used predicatively take the definite article when modified by an of-phrase, though there may be several sons and daughters in the family.

Lomonosov was the son of a fisherman.

Becky Sharp was the daughter of an artist.