Ryze's abilities scale with AP REALLY badly. They do however scale EXTREMELY well with mana. What are the two best mana items in the game? Manume & archangels. And since ryze is burst oriented the archangels and manume will stack mana at roughly the same rate... except that the manume is cheaper... and doesn't give you and useless AP.
So after that we build another tear for even moar mana and then what next? There arn't really any other mana items that don't cost way too much as they provide other bonuses. So why not go for some ad. Trinity force will give you movement AND a on-hit slow as well which means you will suddenly combined with your boots and a snare become and EXCELLENT chaser. Trinity even gives you a small mana boost and since you have about 200 AD mainly thanks you your manume it even has a REALLY nice passive for you.
So then of course we go frozen heart. AD ryze really likes the extra armour and the mana & CDR are obviously fantastic for him. The game will usually end about this time anyway... and compared to any other ryze this ryze will have more mana AND move faster. Because Manume + tear are just that good. If it does continue then a black cleaver is going to start you doing fairly beasty amounts of AD, especially when you start using you passive on trinity aswell. Finally we finish of the archangels is we get far enough... a little *cough +160 cough* AP never hurt anyone did it? This ryze will actually VERY nearly do as much damage as AP/mana stacking ryzes and on top of that he is an AMAZING chaser, can actually escape and can really scare your opponents with a pretty fantastic amount of AD.
He is a bit fragile and if that worries you you could go for a bloodthirster instead of the black cleaver, +100 AD isn't as good as +55 with -60 armour reduction AND +30% AS but the lifesteal is really nice.
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