Subring Test

Subring Test

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 (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 the initial definition (which is used in this article), the only ideal of R that is a subring of R is R itself.

A subring of a ring (R, +, *) is a subgroup of (R, +) which contains the multiplicative identity and is closed under multiplication.

For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z.

The ring Z and its quotients Z/nZ have no subrings (with multiplicative identity) other than the full ring.

Every ring has a unique smallest subring, isomorphic to either the integers Z or some ring Z/nZ with n a nonnegative integer (see characteristic).

The subring test states that for any ring R, a subset of R is a subring if it contains the multiplicative identity of R and is closed under subtraction and multiplication.