A Remarkable Class of Rings

We consider a remarkable class of rings, which we call corpids, that is the rings (K, +, ·), such that (K, ·) is an inverse semigroup (or groupid, which is the name used by Tallini (Ann Math 71:295–322, 1966). We prove several theorems concerning this structure, an order relation which allows us to formulate characterization theorems. We define the notion of simple idempotent and prove theorems about zero-divisors, idempotents, subcorpids, ideals and characteristic in a corpid.