Subring

In mathematics, a subring of a ring R is a subset of R 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.^[a]

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).

Equivalently, S is a subring if and only if it contains the multiplicative identity of R, and is closed under multiplication and subtraction. This is sometimes known as the subring test.^[1]

Some mathematicians define rings without requiring the existence of a multiplicative identity (see Ring (mathematics) § History). In this case, a subring of R is a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). 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 R. With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of R that is a subring of R is R itself.

• The ring of integers ${\displaystyle \mathbb {Z} }$ is a subring of both the field of real numbers and the polynomial ring ${\displaystyle \mathbb {Z} [X]}$.^[1]

• ${\displaystyle \mathbb {Z} }$ and its quotients ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ have no subrings (with multiplicative identity) other than the full ring.^[1]

• 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 ${\displaystyle \mathbb {Z} /0\mathbb {Z} }$.^[2]

• The center of a ring R is a subring of R, and R 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.^{[citation needed]} Since these rings are also real algebras represented by square matrices, the subrings can be identified as subalgebras.

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.^[3] The subring generated by X is also the set of all linear combinations with integer coefficients of elements of X, including the additive identity ("empty combination") and multiplicative identity ("empty product").^{[citation needed]}

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.

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^[b] of S.

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].^[4][3]

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 ${\displaystyle \mathbb {Z} }$.^[3]

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.

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

• Adamson, Iain T. (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3.
• Sharpe, David (1987). Rings and factorization. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.