SQL: how to select last purchased product per client?

Consider the following tables structure:

• invoice: id, client_id
• items: id, invoice_id, name, date

My goal is to get the following dataset: client_id, name, date but the name and date are the columns that correspond to the last purchased product.

I tried the following query, but it does not produce the correct result

SELECT 
    i.client_id, ii.name, MAX(ii.date) 
FROM
    invoice i
INNER JOIN
    invoice_item ii ON i.id = ii.invoice_id
GROUP BY
    i.client_id;

You didn't mention what RDBMS you're using - if it does support CTE's (Common Table Expressions) and windowing functions (many of the most used RDBMS do support it), you could use something like this:

WITH Purchases AS
(
	SELECT 
		i.client_id, ii.name, ii.date,
		RowNum = ROW_NUMBER() OVER (PARTITION BY i.client_id ORDER BY ii.date DESC)
	FROM
		invoice i
	INNER JOIN
		invoice_item ii ON i.id = ii.invoice_id
)
SELECT
	p.client_id, p.name, p.date
FROM
	Purchases
WHERE
	p.RowNum = 1;

The Purchases is a CTE - sort of an "one-off, inline view"; using the ROW_NUMBER function, you "partition" your data by client_id, and all purchases for each client are sequentially numbered, ordered descendingly by purchase date - so the most recent purchase for each client will always have RowNum = 1 - and that's exactly what I select from that CTE.