Ryze objective is to maximize his mana and survivability. Early on he needs some health and mana regen. AP is a secondary concern at best. His most used spell as a 0.2 AP ratio, and scales with 10% of his mana. This should tell you something.
Good items for Ryze:
* Glacial Shroud - mana, armor, and needed cooldown reduction.
* Rod of ages - LOTS of mana, very high health boost, less needed AP. very efficient.
* Tear - Large amount of mana. Ironically, Archangels doesn't really help us as much for its cost. It adds MP5 and AP which we don't need so much, so you only take AA after the 600 cap is hit.
* Banshees - health, mana boost, plus the magic resist. The best anti-magic item for Ryze.
* Sorc Shoes - You dont need Ionic for CDR and this heavily boosts your damage.
I start with a Sapphire and 2 health pots. This provides +20 dmg to his Q, some extra spells before OOM, and lets him regen health decently.
You want a CATALYST as soon as possible. This will prob take 2 backs unless you get a lucky kill which is unlikely esp as you want BOOTS (and a Ruby Crystal hopefully) on your first B. Catalyst is fantastic for Ryze. It provides huge health regen and mana regen. With 250 mana every levelup (which is every 2 minutes or so?) it is more than equal to a Tear. The health regen is very nice too and provides a good toughness boost.
THEN, get a Tear. One would think the Tear first right? No! A catalyst will make you tougher and regen a LOT more health and mana than a tear does. With a catalyst AND tear, your mana will last and last, which really lets you go off and boost your max mana.
Upgrade to Sorc Shoes for the magic pen, which is a large damage upgrade.
Now, go ahead and upgrade the Catalyst to a Rod of Ages. Rod provides over 600 health and 700 mana which is a TON.
To enhance our survivability, get a Negatron and Chain mail now. This plus Rod health makes you very tough.
Upgrade the Chain to a Glacial Shroud. The 15% CDR, plus the 10% from your Q, plus 15% from 9-0-21 and CDR runes maxes you out. This turns you into a machine gun relatively early. Plus Shroud adds even more mana.
Upgrade to Banshee for even more health and mana, plus spell block.
You are pretty much set. You can upgrade to Frozen Heart if you need the extra armor and anti-AD.
Upgrade to Archangels will add some AP for your bouncy and Prison and boost your mana a bit more but it is low priority.
You also have another slot to use as needed for maybe a Guardian's or whatever else you might want.