On the limit closure of domains

**Abstract**

This is a new exposition of an old theorem of Kennison's that
characterizes the limit closure of integral domains inside the category
of commutative rings. A ring in the limit closure must obviously be a
subobject of a product of domains (they are called semi-prime rings and
characterized by the absence of non-zero nilpotents), but not all such
rings are in the limit closure. Rings in the limit closure are
characterized by the fact that any element (1) that is a square mod
every prime and (2) whose cube is a square, is iself a square. They are
also characterized as global sections of a sheaf of domains. For any
commutative ring *R*, the ring of global sections of a certain sheaf of
integral domains is the coreflector of *R* into the limit closure of
domains.