Ho comprato un quaderno a righe per gli appunti di grammatica.

Ho comprato is the passato prossimo, the default past tense for completed actions in everyday Italian.
– Comprato on its own is just a past participle and must be paired with an auxiliary (avere or essere).
– Comprai is the passato remoto, mostly found in literary or very formal contexts, and refers to events in a distant past. In conversation or writing about recent purchases, you’ll almost always use ho comprato.

Italian uses two forms of the masculine singular indefinite article: un and uno.
– Un appears before most consonants and all vowels (e.g. un libro, un amico).
– Uno appears only before s+consonant, z, ps, gn, x, y (e.g. uno studente, uno zaino).
Since quaderno begins with a simple consonant sound /k/ (written “qu-”), it takes un.

A righe means “lined” (literally “with lines”). In Italian, when describing a pattern or characteristic, you often use a + noun:
– a pois = “polka-dotted”
– a quadretti = “checkered”
So a righe is the idiomatic way to describe a notebook that has horizontal lines.

Yes. Rigato is the adjective meaning “lined” or “ruled”, so un quaderno rigato is correct. However:
– a righe is more common in everyday speech.
– rigato can sound slightly more formal or technical (e.g. in stationery catalogs).

Here per expresses purpose: “for” or “in order to.”
– per gli appunti di grammatica = “for (taking) grammar notes.”
It explains why you bought the notebook—i.e. the intended use of the object.

In Italian, to show the subject or content of notes, lectures, etc., you use di.
– appunti di grammatica = “notes of grammar”, meaning “grammar notes.”
Using another preposition (like su or da) would be incorrect here.

When a masculine plural noun begins with a vowel (or certain consonant clusters), the definite article i becomes gli for easier pronunciation:
– gli amici, gli studenti, gli zaini.
Since appunti starts with the vowel A, it takes gli.

Yes, you can drop gli:
– per appunti di grammatica (indefinite/general “for grammar notes”).
Including gli makes it more specific or definite (“for the grammar notes I have/have in mind”). Both forms are grammatically correct; your choice depends on whether you mean notes in general or particular notes.