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Your math is correct. The problem is that most items you are mentioning really shouldn't be rings. Rings generally provide a constant effect onto the wearer (basically the equivalent of range:personal).
Items which cast offensive spells are usually scrolls, wands, rods, or staves. All of which are either uses per day, or number if charges.
I'm not saying it can't be done, just that it's an odd choice, would likely merit increased arbitration by your DM.
Just a couple notes.
You know you can only wear two rings at a time, right? So if you end up making all of these, you will have to decide which two to wear and put the rest in your backpack, or wherever you store them.
I don't think "Use Activated" is what you want in this case. "Use Activated" means you actually use the item to activate the item's power. A use-activated sword power is used when you hit something with the sword. A use-activated potion is used when you drink the potion. Etc.
Consider The One Ring from the Lord of the Rings. It was (among other things) a ring of Invisibility. And in that story, it was "use-activated". Whenever Bilbo or Frodo or Gollum put the ring on their finger, they turned invisible. They "used" the ring by wearing it. When they stopped using it (by removing it) they became visible. This is an example of a "use-activated" ring of invisibility.
Or, consider a more D&D flavored ring of Invisibility. You put the ring on and it sits there on your finger. When you want to turn invisible, you activate your ring. Note, not "use-activate", but simply "activate". Normally in these kinds of games, that means using a command word, but other solutions might be viable, such as twisting the ring around on your finger. However you work this, we call it "Command-Activated".
Do you really want your Scorching Rays to begin shooting out of your fingertips the moment you put on the ring, then keep firing out of your fingers until you take the ring off? If that's what you really want, then "use-activated" would be the right solution.
Note that with this solution, using the ring would require a move action to get it out of your pouch, a move action to put it on, and a move action to take it off. And it would fire continuously the whole time you wear it (maybe even if you're unconscious, which could be very bad) or until you (or someone else) takes a move action to remove it.
But if you want your Ring of Scorching Rays to sit there quietly on your finger, not burning up your home and village, and then when the time is right you activate the ring to scorch your enemies, then "command-activated" is the right solution.
And the good news is, by using this solution, you can save 600 GP and make the item one day faster than making the deadly and problematic use-activated version.
And finally, you cannot create Scorching Ray at Caster Level 2:
A creator can create an item at a lower caster level than her own, but never lower than the minimum level needed to cast the needed spell.
Since a wizard must be level 3 to cast Scorching Ray, the Caster Level of a Ring of Scorching Ray must be at least level 3. It's a small difference, but it's worth noting.
So SL 2 x CL 3 x 1,800 GP = 10,800 GP Base Price for this ring. You'll need 5,400 gp up front and 11 days to make the ring.
Also note, Vampiric Touch and Fireball both require Caster Level 5 when you put them into an item.
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You've got unlimited use cantrips already for your *swing the sword* effect :p Fire of some rays of frost, or acid splash, whatever works for you.
Personally, I restrict my players to items that are written up in the pathfinder core rulebook.
The magic item creation section notes that the formulas are guidelines only, and should be compared to existing items to estimate prices.
A ring of scorching ray, aside from being a standard action to activate, is a 2nd level spell, which means a caster level of 3rd. Note that the minimun caster level for a spell is the minimum level at which you can cast the spell and craft the item. That's 3rd level for a 2nd lvl spell.
So you're looking at:
2*3*1800 (spell level times caster level times command word)
10800g /2
5400g to make
use activation is different from firing a ring. here's the relevant text from the book.
Use Activated: This type of item simply has to be used in
order to activate it. A character has to drink a potion, swing
a sword, interpose a shield to def lect a blow in combat, look
through a lens, sprinkle dust, wear a ring, or don a hat. Use
activation is generally straightforward and self-explanatory.
Also, using a ring to cast a spell isn't quite the same thing as casting the spell yourself. When used through the ring, you don't make an attack bonus yourself. The ring has its own bab that it uses to activate the spell. Check the ring of the ram.
The ring's caster level is its base attack bonus. So, with the above ring, you'd be making your attack roll with a total of +3 to add to the d20 roll. Also, you won't be able to apply any of YOUR feats or abilities to the spell, either metamagic or combat, so no empowered unless you build the ring to use it constantly, and no improved critical, no point blank shot, nothing else you might have either. Which means, at least, an additional -4 to hit when casting it into melee combat.
It's not really a combat viable concept. This is why most items that cast spells either use area spells, or self-buffing spells.
It looks to me more like you're just not happy with the amount of damage that your 1d3 ray of frost is doing, so you're looking for better damage without expending spell slots. You can always take arcane strike and use a melee weapon. But wizards shine as battlefield controllers. Cast Black Tentacles on the next group of bad guys you encounter, and giggle yourself to the bank :)
I seem to have ninja'd someone by 36 minutes. I wonder if I'm on ignore...
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"Use Activated" can be a bit of a fuzzy description. Is the "use" putting the ring on, twisting the ring on the finger, pointing your finger at the enemy, or just thinking "burninate"? Who knows?
Maybe it's just a difference of opinion, but I don't get why you have such an aversion to wands. The fighter needs a sword to slice his enemies, the archer needs a bow, so how it is any less "magical" for you to command flames to burst forth from a twig in your hand? You could take it one step further thematically, and think of the wand like a component of casting the spell, ala Harry Potter and his "swish and flick". Wand mechanics are already in place, no grey areas, there are even premade wands that do what you are looking for.
In fact, if you think of items as being extensions of the components of a spell, then things like Necklaces of Fireballs become options too.
"Use Activated" can be a bit of a fuzzy description. Is the "use" putting the ring on, twisting the ring on the finger, pointing your finger at the enemy, or just thinking "burninate"? Who knows?
Maybe it's just a difference of opinion, but I don't get why you have such an aversion to wands. The fighter needs a sword to slice his enemies, the archer needs a bow, so how it is any less "magical" for you to command flames to burst forth from a twig in your hand? You could take it one step further thematically, and think of the wand like a component of casting the spell, ala Harry Potter and his "swish and flick". Wand mechanics are already in place, no grey areas, there are even premade wands that do what you are looking for.
In fact, if you think of items as being extensions of the components of a spell, then things like Necklaces of Fireballs become options too.
Well, in his defense, having rings do his dirty work kills two birds with one stone:
1) No charges to deal with. It's at-will (requiring a standard action.)2) Doesn't require you to hold something in your hand. So effectively, you can still wield a weapon and a shield and fling spells from your rings.
Okay So if I wanted to make ring of fire balls or lightning bolts but have it do a lower set of dice instead of 5d6 of a lvl 5 caster, how would I do go about scaling a ring thus?
Technically, you can't. Not by RAW.
Alternatively, your DM might allow you, for the required price (Fireball = 3 x 5 x 1,800) to make it at CL 5 (minimum) or even lower if you want, but without reducing the construction price. So for 27,000 GP you could make a ring that casts 1d6 fireballs if you want. Or 5d6, or anywhere in between. I would allow that.
Or, alternatively, your DM might allow you to choose any caster level equal to your own level or lower, and make your ring accordingly, including pricing it based on the caster level. So a 1d6 ring of fireballs would be 3 x 1 x 1,800 = 5,400 gp. I doubt I would allow that (it seems to me that doing this could end up being game-breaking if somebody finds the right combination of a spell and low Caster Level that I didn't foresee). At the very least, I would take the disallow stance and let my players talk me into it on a case-by-case basis (the alternative being to allow it and then maybe have to veto when somebody finds a spell/CL combo that is game-breaking runs into too many arguments).