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Rings
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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
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]
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 of integers is a subring of both the field of real numbers and the polynomial ring .[1]
![{\displaystyle \mathbb {Z} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a538d203a057d4c604f799c28e9a7be410fdcac)
- 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]
- The center of a ring R is a subring of R, and R is an associative algebra over its center.
Subring generated by a set
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 products of elements of X, including the additive identity ("empty combination") and multiplicative identity ("empty product").[4]
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.
Ring extension
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
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, ..., an].[5][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]
![{\displaystyle \mathbb {Z} [i]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffa94e9e2e6d9e5e5373d5fafb954b902743fde)
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 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
Notes
- ↑ In general, not all subsets of a ring R are rings.
- ↑ Not to be confused with the ring-theoretic analog of a group extension.
References
- 1 2 3 Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. p. 228. ISBN 0-471-43334-9.
- ↑ Lang, Serge (2002). Algebra (3 ed.). New York. pp. 89–90. ISBN 978-0387953854.
{{cite book}}: CS1 maint: location missing publisher (link) - 1 2 3 Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913.
- ↑ Robinson, Derek J. S. (2022). Abstract Algebra: An Introduction with Applications (3rd ed.). Walter de Gruyter GmbH & Co KG. p. 109. ISBN 9783110691160.
- ↑ 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
- 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.