# 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 (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.

Read more about Subring:  Subring Generated By A Set, Relation To Ideals, Profile By Commutative Subrings

### Other articles related to "subring, subrings":

Integral Element - Equivalent Definitions
... See also Integrally closed domain Let B be a ring, and let A be a subring of B ... are equivalent (i) b is integral over A (ii) the subring A of B generated by A and b is a finitely generated A-module (iii) there exists a subring C of B containing A and which is a finitely-generated A-mo ... the set of elements of B that are integral over A forms a subring of B containing A ...
Subring Test - Subring Generated By A Set
... Any intersection of subrings of R is again a subring of R ... Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R ... S is the smallest subring of R containing X ...
Profile By Commutative Subrings
... A ring may be profiled by the variety of commutative subrings that it hosts The quaternion ring H contains only the complex plane as a planar subring The coquaternion ring ... group can be realized as the join of the groups of units of two of these nilpotent-generated subrings of 3 × 3 matrices ...