What if a genasi wore this? Would they become immune?

Nope.  Resistances don't stack to become immunity, so if you have resistance from more than one source, you don't gain any real benefit.

CassieCardoc is right, but your DM might allow resistances to stack so the normal half damage becomes a quarter damage, but always does at least one.

Though I would think it better to spread the resistance to as many party members as possible as opposed to making a super tank for one specific type of damage. 

 A way my DM plays with this is he makes it so these kinds of items reduce types of damage by half and then your resistance kicks in

A wise purchase, thank you Fjord.

We just did a massive quest for my Treant Bard to acquire one of this rings so that he wouldn't just take normal damage from fire attacks. 

Just watched that episode lmao

Imagine getting all ten rings of resistance and being an unkillable juggernaut 

no, the damage would be quartered