Betta tails

Question
I recentally bought a blue mystery snail to put in Divinci's (my betta) tank.  I've had several snails in there before with no problem whatsoever.  I even gone through a pond snail infestation with no problems.  Even since I got my current snail, Divinci's tail seems to be...for lack of a better description, dissolving.  I thought he may have tail rot when I first saw the hole in his tail, but then I realized that it looked like a burn mark in the middle of his tail, it's slowly grown from there.  It has black edges that looks like a burn mark.

My tank setup: dirt, gravel, and anubis plant. no filter, and one gallon glass container.

Answer
I think that you need to get a filter for your betta.  whether it is fin and tail rot or a chemical burn, the filter is most likely the answer.  you can get tiny filters for betta bowls, etc at the local aquarium store for under $20 and they work off of an air pump (included in the $20)... dave