If you were actually going for historic accuracy, or planned to use it for armor then you would weld the rings. Generally people don't weld them since it takes a lot of time and effort. They are just bent closed.

Historically it would be riveted not welded. Historical example

Correct. Occasionally in my darker moments I've actually considered making a set with riveted rings. Every time I've ultimately come to my senses and concluded that I can't be bothered.

You'll notice that authentic mail usually uses rings that are flat in cross section, too, what with medieval Celts or whoever not having access to unlimited amounts of our modern machine made wire.

I can feel the amount of time and effort on display in that photo.