Let
R be a
ring (with 1). A subset
S of
R
is a
subring of
R if it is itself a ring with the
operations
and
identity of
R.
Equivalently, a subset S of R is a subring if and only if

1_{R} in S

ab in S, for all a,b in S

ab in S, for all a,b in S
For example, the ring of integers is a subring of the ring of
rational numbers. The collection of upper triangular 2x2 matrices
with complex coefficients is a subring of the ring of 2x2 complex
matrices.