8mm Mauser = common name for what is actually properly termed
8x57IS which is the same as
8x57JS (Germanoften uses a J for a capital i to avoid confusion with a 1)
which is also often called
7.92x57 abroad. As you will shortly see, the 7.92 is incorrect.
BUT the dangerous confusion is that the original Gew88 used a cartridge designated 8x57I (or 8x57J) , no S.
I stood for Infantry.
The bore/groove dimensions were 7.80/8.07mm. Please note NOT 7.92.
This was upgraded to take a "Spitzer" bullet, hence the S.
The problem was, that the S bullet required larger bore/groove dimensions
namely 7.89/8.20mm. Please note once again NOT 7.92.
There was indeed a cartridge "7.92x33 kurz", used for the "Sturmgewehre" /machine pistols.
Firing S-type .323" / 8.20mm bullets down an original I-bore with 8.07mm groove diameter is therefore NOT A GOOD IDEA.
However, there are very few I-bores left, as the vast majority of Gew88s were converted from 1905 on, The adaptation is indicated by an S stamped on the barrel ring, towards the rear sight.
As to the designations with R: it's the same again, but the R simply indicates a rimmed cartridge. Rimmed 8x57IRS cartridges are quite common for hunting rifles. Once again, you are very unlikely to come across an IR type, unless you collect antique German hunting rifles.
Regardless of what anyone's micrometer might read. all the above dimensions are the official standard values from the "Masstafeln" of 10. Jan. 2000. Worn bores will, of course, show higher figures.
Patrick