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.^[1]^: 228

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 $\mathbb {Z} /0\mathbb {Z}$ .^[2]^: 89–90

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.^[1]^: 228

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 $\mathbb {Z}$ 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 $0$ .

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).

References

Adamson, Iain T. (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3.
Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. ISBN 0-471-43334-9.
Lang, Serge (2002). Algebra (3 ed.). New York. ISBN 978-0387953854.{{cite book}}: CS1 maint: location missing publisher (link)
Sharpe, David (1987). Rings and factorization. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.

[FOOTNOTEDummitFoote2004-1] Jump up to: ^a ^b Dummit & Foote 2004.

[FOOTNOTELang2002-2] Lang 2002.

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