OK, so this is not earth shattering, but it can come in handy. We used to do this in the flower shop as a way to clean up vases. All you need is some Denture Tablets and you vase with some water.
Here the tablet is in the water, fizzing up. This is a fun thing for your kids to watch too! If you have a pretty large vase, you might need to add 2 tablets. You can get a box of these denture tablets at any grocery store or pharmacy. Get the generic kind to save money and this box will last you forever!
I am doing a few vases here at a time. I just let them sit overnight...
Now all you have to do is dump the water and take a paper towel to wipe out that ring and it really clean! Give it a quick rinse of clean water and let the vase dry upside down. Now they are clean and ready for that next bunch of fresh flowers!