Subring

In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all subsets of a ring are rings.

Definition

A subring of a ring is a subset of that preserves the structure of the ring, i.e. a ring with . Equivalently, it is both a subgroup of and a submonoid of .

Equivalently, is a subring if and only if it contains the multiplicative identity of , and is closed under multiplication and subtraction. This is sometimes known as the ''subring test''.

Variations

Some mathematicians define rings without requiring the existence of a multiplicative identity (see '). In this case, a subring of is a subset of that is a ring for the operations of (this does imply it contains the additive identity of ). This alternate definition gives a strictly weaker condition, even for rings that do have a multiplicative identity, in that all ideals become subrings, and they may have a multiplicative identity that differs from the one of . With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of that is a subring of is itself.

Examples

* The ring of integers $\Z$ is a subring of both the field of real numbers and the polynomial ring \Z /math>.

* $\mathbb$ and its quotients $\mathbb/n\mathbb$ have no subrings (with multiplicative identity) other than the full ring.

* Every ring has a unique smallest subring, isomorphic to some ring $\mathbb/n\mathbb$ with ''n'' a nonnegative integer (see '' Characteristic''). The integers $\mathbb$ correspond to in this statement, since $\mathbb$ is isomorphic to $\mathbb/0\mathbb$.

* The center of a ring is a subring of , and is an associative algebra over its center.

Subring generated by a set

A special kind of subring of a ring is the subring generated by a subset , which is defined as the intersection of all subrings of containing . The subring generated by is also the set of all linear combinations with integer coefficients of products of elements of , including the additive identity ("empty combination") and multiplicative identity ("empty product").

Any intersection of subrings of is itself a subring of ; therefore, the subring generated by (denoted here as ) is indeed a subring of . This subring is the smallest subring of containing ; that is, if is any other subring of containing , then .

Since itself is a subring of , if is generated by , it is said that the ring is ''generated by'' .

Ring extension

Subrings generalize some aspects of field extensions. If is a subring of a ring , then equivalently is said to be a ring extensionNot to be confused with the ring-theoretic analog of a group extension. of .

Adjoining

If is a ring and is a subring of generated by , where is a subring, then is a ring extension and is said to be ''adjoined to'' , denoted . Individual elements can also be adjoined to a subring, denoted .

For example, the ring of Gaussian integers \Z /math> is a subring of $\C$ generated by $\Z \cup \$, and thus is the adjunction of the imaginary unit to $\Z$.

Prime subring

The intersection of all subrings of a ring is a subring that may be called the ''prime subring'' of by analogy with prime fields.

The prime subring of a ring is a subring of the center of , which is isomorphic either to the ring $\Z$ of the integers or to the ring of the integers modulo , where is the smallest positive integer such that the sum of copies of equals .

See also

* Integral extension
* Group extension
* Algebraic extension
* Ore extension

Notes

References

General references

*
* {{cite book , last1=Sharpe , first1=David , title=Rings and factorization , url=https://archive.org/details/ringsfactorizati0000shar , url-access=registration , publisher=Cambridge University Press , date=1987 , isbn=0-521-33718-6 , page
15–17
}

Ring theory