Subring

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 that shares the same multiplicative identity as R. (Note that a subset of a ring R need not be a ring.) 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 definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself.

Definition
A subring of a ring (R, +, ∗, 0, 1) is a subset S of R 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).

Examples
The ring $$\mathbb{Z}$$ 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 n = 0 in this statement, since $$\mathbb{Z}$$ is isomorphic to $$\mathbb{Z}/0\mathbb{Z}$$.

The ring of split-quaternions has subrings isomorphic to the rings of dual numbers, split-complex numbers and to the complex number field.

Subring test
The subring test is a theorem that states that for any ring R, a subset S of R is a subring if and only if it contains the multiplicative identity of R, and is closed under multiplication and subtraction.

As an 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[X].

Center
The center of a ring is the set of the elements of the ring that commute with every other element of the ring. That is, $x$ belongs to the center of the ring $R$ if $$xy=yx$$ for every $$y\in R.$$

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

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 $$\Z$$ 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 $0$.

Ring extensions
If S is a subring of a ring R, then equivalently R is said to be a ring extension of S.

Subring generated by a set
Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. This subring is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.

The subring generated by X is the set of all linear combinations with integer coefficients of products of elements of X (including the empty linear combination, which is 0, and the empty product, which is 1).