This build may look awkward to you at first glance but look tear gives you mana everytime you cast as well as everytime you lvl you get more base mana and your 3 quints giving (12.51) mana per level. It doesnt seem like much but you will have 225 mana just from your quints alone plus the 101 mana from your seals= to 326 mana extra end game not including your base mana plus your tears mana.

Ionian Boots of Lucidity Boots for the cooldown of course more cooldown red.= more damage output.

Why Banshee's veil you ask? Let me tell you, as ryze you are are a squishy caster which means the more protection you have the better, Banshee's Veil give you +375 Health +375 Mana +50 Magic Resistance, and also Blocks one negative spell every 45 seconds. This helps with Karthus ult as well as Ash'es arrow even Twisted fates ult. And with pretty much and spell/ability that single targets you.

Rod of Ages is a must, why? 450 Health +525 Mana +60 Ability Power plus a additional +180 Health, +200 Mana, and +20 Ability Power. Add that all up and that makes 630 health 725 mana and 80 ability power(But i dont need ability power im ryze) Wrong Your spell flux is still based off of your Ability Power.

Frozen Heart +99 Armor and +500 mana as well as Reduces ability cooldowns by 20% and reduces the attack speed of nearby enemies by 20%. Cooldown reduction works reall well with Ryze's passive
Void Staff of course for the magic pen as well as the Ability power.

And the Archangel's Staffjust for extra mana and mana regen.

Try not to auto attack to much just last hit the creeps, if your mana is looking ok throw a few spells to get more mana from your tear.

At the end of the game you should have around 5k mana if not more, you will be able to do sick burst damage and even out dps a carry. Good luck with my build hope it helped :)