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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.)

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).

Variations

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.

Examples

The ring ${\displaystyle \mathbb {Z} }$ 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 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.

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 ${\displaystyle xy=yx}$ for every ${\displaystyle y\in R.}$

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 ${\displaystyle \mathbb {Z} }$ 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.

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).

See also

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

Notes

1. ^ Dummit & Foote 2004.
2. Lang 2002.

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.

• Ring theory