For any reader here who doesn't already know:
The proper name for that object is a cartridge-casing. It is the "necked" variety of casing, meaning, its upper body is significantly smaller in diameter than its lower body. "Necked" casings first came into official use in the 1870s, and are the standard form for rifle cartridge-casings up through today.
Digging1971guy, to identify your mystery-casing, we'll need precise diameter measurements (taken with digital calipers), at the casing's top, and just above the wide rim at its base, and also the casing's length. I realize it is damaged, but please do the best you can at getting accurate measurements.