Hey Guys!

Sometimes I'm rushing Tear of the Goddess, sometimes Catalyst or even T2-Boots.
This choice always depends on my lane opponent.
I try to rush Tear of the Goddess, but in my opinion it's not always the right choice.

I feel relatively safe and confidant that I can escape gank attempts? Tear of the Goddess
I need to counter push and sustain myself (e.g. against Morgana)? Catalyst
I need to dodge nasty skill shots? T2-boots
I'm playing solo top vs. an AD bruiser? Glacial Shroud

I consider Catalyst, Tear and Glacial Shroud as my core.
I pick different boots depending on the game.
Do I need to tank for my team? Ninja Tabi or Mercury Treads
Do they've got lots of CC? Mercury Treads
Do they've got one or multiple scary AD-champs? Ninja Tabi
If I answer all these question with no, I pick Sorcerer's Shoes.

You need more survivability? Will of the Ancients or MR/Armor, e.g. Guardian Angel, Abyssal Scepter
You need more damage and they've built MR? Void Staff
You need more damage and they haven't built MR? get 40% CDR
(after you've got Tear, Catalyst and Frozen Heart, sadly there aren't any great mana items left, but usually you'll end up building Void Staff or WotA anyway.)

Rod of Ages is a very strong item on Ryze, because it renders him very tanky, while still increasing his damage. But I rejected it, because I don't find room for it and it delays my Tear and/or Glacial Shroud. (You really don't want to delay Rod of Ages, if you plan to build it.)

BV is needed because there are only few MR choices for Ryze.
WotA is great, because Spellvamp is godlike on Ryze.
Void Staff is needed to keep your damage as high as possible and to allow you to build Merc Treads or Ninja Tabi.

If I really need to be the main tank/initiator and I know it right from the beginning, I'd probably consider to rush a Rod of Ages, but otherwise I consider the build above superior.

I usually don't finish Archangel Staff, because most of your damage comes from your Q, which scales badly with AP. You spend a lot of money for a minor damage increase. I usually prefer to get more tanky to be able to cast longer during fights.

That said, there are a lot of possible good item choice for Ryze and I'm sure different item paths are viable too.

Exhaust: Opponents movement speed is reduced, attack damage dealt..reduced, ability and item damage...reduced. This is perfect for Ryze to have in his arsenal as you're gonna be cranking out huge damage getting your OVERLOAD off 1st then followed up by RUNE PRISON, which by the time that wears off, should be able to finish with another RUNE PRISON if needed. This combo is all about SLOW them down and hitting harder.

Teleport: I chose this because you never know when an ally will need help. Being a HUGE damage dealer you can pop in and provide some serious cover fire.

PROS
Heavy Damage output end game
You have Spell Vamp and all you do is cast mana, great for self healing
Quick CDR's with build listed
2 skills give you AoE's
2 single skills work based off max mana % for bonus damage...this build gives you the powder keg

CONS
You're gonna be squishy to start...harass opposing champions but focus on minions
Not as many AoE's as some other casters
team might call you a KSer for 1-2 hitting a Champion they were attacking...but thats the price you pay for Ryze Mayhem

Start with:

Tear of the Goddess + Boots of Speed
or
Catalyst the Protector + Health Potion

Catalyst the Protector is also a good starting item because it gives you both mana and health and restores them when you level up, BUT ...
Starting with Tear of the Goddess will help you stack more mana in the end and you have money left for the Boots of Speed.
We aim for large amounts of mana so I choose Tear of the Goddess.

After:

1. Get Catalyst the Protector if you started with Tear of the Goddess or the other way round. Mana Crystal if you die early.
2. Upgrade to Sorcerer's Shoes
3. Glacial Shroud ( You should have 34% Cooldown Reduction now )

In the end:

If you need Magic Resist fast, build Odyn's Veil from Catalyst the Protector
If you also need Armor, finish Frozen Heart faster, else leave it for later.
If your Magic Resist can wait (you are doing well/ the other team is AD) build Rod of Ages from Catalyst the Protector early ( will increase mana by minute = more damage later ) and get Odyn's Veil later.

Hextech Revolver -> Will of the Ancients will increase Spell Vamp, your Ulti Desperate Power does that too, so you will heal yourself large amounts in fights!
upgrade your Tear of the Goddess to Archangel's Staff for 50 extra mana and some good damage!

Let's say you have 3000 mana, Archangel's Staff will give you 3% of this ( 90 AP ) + 45 for 1860 GOLD !!!
135 AP means +27 DAMAGE ON (Q) Overload (20%), +81 DAMAGE ON (W) Rune Trap (60%), +47.25 DAMAGE ON (E) Spell Flux (35%)
( +155.25 overall damage increase )

Now let's see Rod of Ages ( 80 AP, 725 Mana, 630 Health ) 10 minutes to maximize.
725 Mana means + 58 DAMAGE ON (Q) Overload (8%) , +36.25 DAMAGE ON (W) Rune Trap (5%) , 0 DAMAGE ON (E) Spell Flux (0%) = 94.25 total from mana,
80 AP gives 16+48+28 = 92 total damage from AP

So Rod of Ages will give you 92+94.25 = 186.25 overall damage + 630 Health