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 (R, +, *, 0, 1) is a subset Template:Mvar of Template:Mvar that preserves the structure of the ring, i.e. a ring (S, +, *, 0, 1) with S ⊆ R. Equivalently, it is both a subgroup of (R, +, 0) and a submonoid of (R, *, 1).

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

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

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

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

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 products of elements of Template:Mvar, including the additive identity ("empty combination") and multiplicative identity ("empty product").

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 S ⊆ T.

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 R ∪ S, 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 R[S]. Individual elements can also be adjoined to a subring, denoted R[a₁, a₂, ..., a_n].

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

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 $\mathbb{\mathbb{Z}}$ 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 1 equals 0.

See also

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