gday p2as2k3, after you have got the emerald from the deku tree go to Zeldas castle which is through the market where Talon is sleeping, (you will have to get the egg from malon at night in the castle path and move the boxes to wake him up) and you will see the drain where the water is coming out, after Talon is gone grab 2 boxes which he was sleeping near and put them in the gap directly behind the drain, then jump on the boxes and crawl through the drain, from here it is just a matter of sneaking past guards, after you've passes them all you will meet Zelda, she will talk to you and give you some sort of permission slip for the guard at the gate of death mountain through Kakaro Village, take the slip and go to the guard at Kakaro Village, give him the slip and he will let you pass into death mountain( I recommend you have a Hyrule shield first) and head to the top of the mountain, you will see a cave at the top, go through it and you will be in Goron City and this is where the RED RUBY RING is............
it is pretty easy from here but you can email me on Janandlinc@Bigpond.com if you have any questions IM THE MAN
cheers MasterOwey01

Um, actually, there's no deku tree in the original Legend of Zelda for NES and the explanation in the above answer also isn't correct for this version of the game.

For this game, the red ring is located inside the 9th dungeon. Here's how to get it, from the starting point of the level:

Up 2 screens.
Bomb and go left 1 screen
Kill the red centipede things and take the secret passage way. (watch out for the Like Likes when you come out).
Right 2 screens
Down 1 screen
Bomb and go right 1 screen
Bomb and go up 2 screens (yep, you need to bomb both times)
Kill all the critters in this room and take the stairs to the red ring.

Good luck!

P.S. Here is a gif map of the level 9 dungeon, just in case. =0)

http://db.gamefaqs.com/console/nes/..._dungeon_09.gif