Definition
Category of unital rings
Unital associative rings and unit-preserving ring homomorphisms.
The category of unital rings has unital associative rings as objects. Morphisms preserve addition, multiplication, and the unit: . Identities and composition are the corresponding functions. The zero ring is allowed as an object.
Convention and comparison
The zero function into a nonzero unital ring is not a morphism. An inclusion of a matrix corner generally belongs to the category with arbitrary ring homomorphisms but not to this category, because it sends the smaller unit to a proper idempotent.