Subring

Summary

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]

Definition

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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 SR. 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]

Variations

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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

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  •   and its quotients   have no subrings (with multiplicative identity) other than the full ring.[1]
  • Every ring has a unique smallest subring, isomorphic to some ring   with n a nonnegative integer (see Characteristic). The integers   correspond to n = 0 in this statement, since   is isomorphic to  .[2]

Subring generated by a set

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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 ST.

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

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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.

Adjoining

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If A is a ring and T is a subring of A generated by RS, 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, ..., an].[4][3]

For example, the ring of Gaussian integers   is a subring of   generated by  , and thus is the adjunction of the imaginary unit i to  .[3]

Prime subring

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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   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.

See also

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Notes

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  1. ^ In general, not all subsets of a ring R are rings.
  2. ^ Not to be confused with the ring-theoretic analog of a group extension.

References

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  1. ^ a b c Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. p. 228. ISBN 0-471-43334-9.
  2. ^ Lang, Serge (2002). Algebra (3 ed.). New York. pp. 89–90. ISBN 978-0387953854.{{cite book}}: CS1 maint: location missing publisher (link)[dead link]
  3. ^ a b c Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913.
  4. ^ Gouvêa, Fernando Q. (2012). "Rings and Modules". A Guide to Groups, Rings, and Fields. Washington, DC: Mathematical Association of America. p. 145. ISBN 9780883853559.

General references

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  • 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.