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 shares the same
multiplicative identity as
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 (this does imply it contains the additive identity 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 (and they may have a multiplicative identity that differs from the one of
R). With definition requiring a multiplicative identity (which is used in this article), the only ideal of
R that is a subring of
R is
R itself.
