Subring

In mathematics, a subring of a ring Template:Mvar is a subset of Template:Mvar 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 Template:Mvar.

Definition

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

Equivalently, Template:Mvar is a subring if and only if it contains the multiplicative identity of Template:Mvar, 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 Template:Slink). In this case, a subring of Template:Mvar is a subset of Template:Mvar that is a ring for the operations of Template:Mvar (this does imply it contains the additive identity of Template:Mvar). 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 Template:Mvar. With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of Template:Mvar that is a subring of Template:Mvar is Template:Mvar itself.

Examples

• The ring of integers $\mathbb{Z}$ is a subring of both the field of real numbers and the polynomial ring $\mathbb{Z}[X]$.

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

• Every ring has a unique smallest subring, isomorphic to some ring $\mathbb{Z}/n\mathbb{Z}$ with n a nonnegative integer (see Characteristic). The integers $\mathbb{Z}$ correspond to Template:Nowrap in this statement, since $\mathbb{Z}$ is isomorphic to $\mathbb{Z}/0\mathbb{Z}$.

• The center of a ring Template:Mvar is a subring of Template:Mvar, and Template:Mvar is an associative algebra over its center.

• The ring of split-quaternions has subrings isomorphic to the rings of dual numbers and split-complex numbers, and to the complex number field.Template:Cn Since these rings are also real algebras represented by square matrices, the subrings can be identified as subalgebras.

Subring generated by a set

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

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

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

Ring extension

Subrings generalize some aspects of field extensions. If Template:Mvar is a subring of a ring Template:Mvar, then equivalently Template:Mvar is said to be a ring extension of Template:Mvar.

Adjoining

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

For example, the ring of Gaussian integers $\mathbb{Z}[i]$ is a subring of $ℂ$ generated by $\mathbb{Z}\cup \{i\}$, and thus is the adjunction of the imaginary unit Template:Mvar to $\mathbb{Z}$.

Prime subring

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

The prime subring of a ring Template:Mvar is a subring of the center of Template:Mvar, which is isomorphic either to the ring $\mathbb{Z}$ of the integers or to the ring of the [[modular arithmetic|integers modulo Template:Mvar]], where Template:Mvar is the smallest positive integer such that the sum of Template:Mvar copies of Template:Math equals Template:Math.

See also

• Integral extension
• Group extension
• Algebraic extension
• Ore extension

References

General references

• Template:Cite book
• Template:Cite book