Subring

Examples

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

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

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

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

Subring generated by a set

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

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

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

Ring extension

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

Adjoining

If A is a ring and T is a subring of A generated by R ∪ S, where R is a subring, then T is a ring extension and is said to be S adjoined to R, 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 ${\displaystyle \mathbb{Z}[i]}$ is a subring of ${\displaystyle \mathbb{C}}$ generated by ${\displaystyle \mathbb{Z}\cup \{i\}}$, and thus is the adjunction of the imaginary unit i to ${\displaystyle \mathbb{Z}}$.

Prime subring

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

The prime subring of a ring R is a subring of the center of R, which is isomorphic either to the ring ${\displaystyle \mathbb{Z}}$ of the integers or to the ring of the integers modulo n, where n is the smallest positive integer such that the sum of n copies of 1 equals 0.