Given a RinG (R,+,*), we say that a subset S of R is a subring thereof if it is a ring under the restriction of + and * thereto. A subring is just a SubGroup of (R,+) closed under multiplication.

A subring with unity is a subring which includes, not just any unity, but the *same* unity as the parent ring. Of course this is the only sort of subring we talk about if we define rings to have a unity by default. Every ring has a unique smallest subring with unity, isomorphic to either the IntegerNumbers or some ModularArithmetic.

Subrings closed under multiplication by *arbitrary* ring elements are called ideals. Clearly the only ideal which contains unity is the ring itself. These play exactly the same role that normal SubGroup*s* do for groups, being precisely the possible kernels of ring HomoMorphism*s*.