User:TMM53/Subrings-2023-03-21

This page is about subrings. For rng, an algebraic structure like a ring but without an identity element, see Rng (algebra).

In mathematics, a ring is a set and 2 binary operations, addition and multiplication with additive and multiplicative identites. A subring arises from a ring's subset by restriction of the ring's addition and multiplication operations to the ring's subset and sharing a common multiplicative identity. This transforms the ring's subset to a subring. The relationship between rings and subrings preserves the ring's structure. This means that for any shared elements of ring and subring, the sum or product of these elements in the subring matches a corresponding sum or product in the ring.

Properties

A ring has the properties of a commutative additive group, and associative multiplication with a multiplicative identity element. Associative multiplication with an identity element means that a ring has the properties of a multiplicative monoid. Therefore, a subring contains an additive subgroup of the ring's additive group and a multiplicative submonoid of the ring's monoid.

A proper subring has a proper subset of the ring's set. An improper subring has an improper subset of the ring's set.

The subring-ring relationship is transitive.

A ring and its subrings may not share identical properties. For example, a Noetherian ring may have a non-Noetherian subring.

Examples

Rings with subrings

• The integers are a subring of the rational numbers, and the rational numbers are a subring of the real numbers.
• The complex numbers are a subring of the quaternions, and the quaternions are a subring of 2 x 2 matrices with complex number entries.
• Every ring has a unique smallest subring isomorphic to a quotient ring ${\textstyle \mathbb {Z} /n\mathbb {Z} }$ with positive integer ${\textstyle n}$, the ring's characteristic, or the ring of integers.

Rings without proper subrings

• The integers and the quotient rings have no proper subrings.

Subring test

Ring ${\textstyle R}$ is a proper subring of ring ${\textstyle S}$ if $R$ is non-empty, shares the same identity element of ring ${\textstyle S}$, is closed under multiplication and subtraction and is a proper subset of ring.${\textstyle S}$.

Ring extensions

Not to be confused with a ring-theoretic analog of a group extension. For that meaning, see an old version of the article Idealization of a module.

Ring $S$ is a ring extension of ring $R$ is equivalent to ring $R$ is a subring of ring $S$. Ring extension notation, $S/R$, is similar to field extension notation.

Subring generated by a set

The intersection of any family of subrings is a subring. The intersection of any family of subrings containing a common set is a subring. The smallest subring containing a common set is the intersection of all subrings containing the common set. Set ${\textstyle A}$ generates ring $S$ in ring $R$ if $S$ is the smallest subring in $R$ containing set ${\textstyle A}$.

See also

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

Category:Ring theory

References

• Iain T. Adamson (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.
• Hartley, B.; Hawkes, T.O. (1974). Rings, modules and linear algebra: a further course in algebra describing the structure of Abelian groups and canonical forms of matrices through the study of rings and modules. London: Chapman & Hall. ISBN 978-0412098109.
• Jacobson, Nathan (1989). Basic algebra (2nd ed.). New York: W.H. Freeman. ISBN 0-7167-1480-9.
• Kuz'min, Leonid Viktorovich (2002). Encyclopaedia of mathematics. Berlin: Springer-Verlag. ISBN 1402006098.
• Lang, Serge (2002). Algebra (3 ed.). New York. ISBN 978-0387953854.{{cite book}}: CS1 maint: location missing publisher (link)}
• Larsen, Max D.; McCarthy, Paul J. (1971). Multiplicative theory of ideals (PDF). New York: Academic Press. ISBN 978-0124368507.
• Rosenfeld, Boris (1997). Geometry of Lie Groups. Boston, MA: Springer US. ISBN 978-1-4419-4769-7.
• David Sharpe (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6.