how to reglue ringmat?

I've owned a ringmat for 7-8 yrs with no problems. All I can say is that IF I had this problem, the only people I'd be asking for advice would be Ringmat! Have you tried contacting them?

Practice spraying or brushing on glue before you do the real thing to get the hang of it! Especially if you have two left thumbs:>)

GLue for paper and cork is rubber cement.
It will hold, and not damage the surfaces. When it too gets old, it will come apart, but can be redone.
If you try a more drastic, stronger glue, and it gets messed up, it may not as easily redo-able.
So I say common rubber cement is the best. Art store stuff.
If not OK, then a spray automotive glue from 3M (this sort of glue comes out thick even though a spray, so application can be problematic)

Or a spray art poster glue..