Subring

Last updated October 30, 2024

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 $R$ .^{[lower-alpha 1]}

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 $S \subseteq R$ . 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 $\mathbb {Z}$ is a subring of both the field of real numbers and the polynomial ring $\mathbb {Z} [X]$ .^[1]

$\mathbb {Z}$ and its quotients $\mathbb {Z} /n\mathbb {Z}$ have no subrings (with multiplicative identity) other than the full ring.^[1]

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]

The center of a ring $R$ is a subring of $R$ , and $R$ is an associative algebra over its center.

The ring of split-quaternions has subrings isomorphic to the rings of dual numbers and split-complex numbers, and to the complex number field.^{[ citation needed ]} Since these rings are also real algebras represented by square matrices, the subrings can be identified as subalgebras.

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 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 $S \subseteq T$ .

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^{[lower-alpha 2]} of $S$ .

Adjoining

If $A$ is a ring and $T$ is a subring of $A$ generated by $R \cup S$ , 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 $R [a 1, a 2, ..., a n]$ .^[4]^[3]

For example, the ring of Gaussian integers $\mathbb {Z} [i]$ is a subring of $\mathbb {C}$ generated by $\mathbb {Z} \cup \{i\}$ , and thus is the adjunction of the imaginary unit $i$ to $\mathbb {Z}$ .^[3]

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 $0$ .

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.

Related Research Articles

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In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.

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In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial whose coefficients are integers. The set of all algebraic integers $A$ is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.

<span class="mw-page-title-main">Generating set of a group</span> Abstract algebra concept

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In algebra, a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.

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In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A.

In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.

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)^{[ dead link ]}
1 2 3 Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913.
↑ 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.

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[1] In general, not all subsets of a ring $R$ are rings.

[5] Not to be confused with the ring-theoretic analog of a group extension.

[Dummit_&_Foote-2] 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.

[3] Lang, Serge (2002). Algebra (3 ed.). New York. pp. 89–90. ISBN 978-0387953854.{{cite book}}: CS1 maint: location missing publisher (link)^{[ dead link ]}

[lovett-4] 1 2 3 Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913.

[6] Gouvêa, Fernando Q. (2012). "Rings and Modules". A Guide to Groups, Rings, and Fields. Washington, DC: Mathematical Association of America. p. 145. ISBN 9780883853559.

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