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So I have the Lesser Ring of Power:
Damage Resistance: cold resist 5 / -
Damage Resistance: electrical resist 5 / -
Damage Resistance: fire resist 5 / -

and the Ring of Elemental Resistance:
Damage Resistance: acid resist 15 / -
Damage Resistance: cold resist 15 / -
Damage Resistance: electrical resist 15 / -
Damage Resistance: fire resist 15 / -

...and when I get shocked from an electric charge it says it only absorbed 15 damage. Does anyone know if these effects stack or not?

They don't - as far as I recall. Only the highest (which also seems to be what you experienced) will be in effect.

Damage resistance typically stacks, but I don't know about elemental damage. Did it say that you took extra electrical damage above what was absorbed? If not, then it may be that only 15 pts were dealt.

If you have multiple damage absorbing items (such as, say, the greater brawler's belt and the adventurer's robe) then you will get a separate line for each item. So, if the enemy did 30 pts of damage there will be 3 lines:

X took 5 pts (5 normal)
X absorbed 20 pts
X absorbe 5 pts

I haven't got the verbiage exactly right, but you get the picture. The point is that that extra line might get lost in the shuffle if there is a lot going on.

But like I said, I know this applies to normal damage but am not so sure about elemental. I think I have seen the multi-line effect with fire damage but I don't really pay much attention since it is usually a small amount and tends to get regenerated before it becomes a problem.

I know that DR/- don't stack with DR/-. A Greater swordsman's belt gives 20/- against slashing. It don't stack with another Item that gives 20/- against slashing. Damage Resistance of the type that can you can overcome stacks with Damage Resistance like the one Greater Swordsman's belt give.