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**/*n***Z** 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**/*n***Z** 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

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