Iḍāfa: the possessive construction
Arabic · Key Stage 3 · The School
Two nouns, one meaning
كِتابُ الوَلَدِ — the boy's book. The first noun (mudāf) never takes ال and never takes tanween; the second (mudāf ilayhi) is in the genitive. Two nouns side by side under those rules automatically mean "X of Y". No word for "of" is needed.
The rule that catches everyone
You cannot say الكِتابُ الوَلَدِ. If the first noun wears ال, it is no longer an iḍāfa. Definiteness comes from the SECOND noun: كِتابُ الوَلَدِ is definite ("the boy's book") even though كِتاب has no article. This surprises every learner exactly once.
Everywhere in the Quran
بِسْمِ اللهِ — "in the name of Allah": ism + Allah, an iḍāfa. رَبِّ العالَمينَ — "Lord of the worlds". يَوْمِ الدّينِ — "the Day of Judgement". Once you see the pattern, the opening of Al-Fatihah reads as a chain of iḍāfas, and its grammar becomes as clear as its meaning.
WRONG: البابُ البَيْتِ — two definite nouns, which is not an iḍāfa at all. RIGHT: بابُ البَيْتِ — the first noun drops ال and takes a ḍamma; the second takes ال and a KASRA. It means 'the door of the house' and the first word is definite because of the second. Now read رَبُّ العالَمين and بِسْمِ اللهِ with that in mind — both are iḍāfa.