Subring
In mathematics, a
subring of
R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on
R are restricted to the subset, and which contains the multiplicative identity of
R. For those who define rings without requiring the existence of a multiplicative identity, a subring of
R is just a subset of
R that is a ring for the operations of
R. The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings. With definition requiring a multiplicative identity, the only ideal of
R that is a subring of
R is
R itself. A subring of a ring is a subset
S of
R that preserves the structure of the ring, i.e. a ring with
S ⊆
R. Equivalently, it is both a subgroup of and a submonoid of. The ring
Z and its quotients
Z/
nZ have no subrings other than the full ring. Every ring has a unique smallest subring, isomorphic to some ring
Z/
nZ with
n a nonnegative integer.