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Algebraic structure → Ring theory
Ring theory
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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

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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 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

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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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See also: Generator (mathematics)

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

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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 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]

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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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Subring

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In ring theory, a subring of a ring RR is a nonempty subset S⊆RS \subseteq R that is closed under the addition and multiplication operations inherited from RR, contains the additive identity 00 of RR, and is closed under additive inverses, thereby forming a ring itself under these operations. A standard test for verifying a subset SS is a subring requires showing that SS is nonempty, closed under subtraction (which ensures closure under addition and additive inverses), and closed under multiplication. Definitions of subrings vary slightly across mathematical , particularly regarding the multiplicative identity. In contexts emphasizing rings with unity (multiplicative identity 11), such as , subrings are often required to contain the same 11 as RR, ensuring compatibility with units and ring homomorphisms. However, in more general treatments of , subrings need not include 1R1_R; for instance, the even integers 2Z2\mathbb{Z} form a subring of the integers Z\mathbb{Z} despite lacking 11. This distinction arises from historical developments in , where early definitions sometimes omitted unity to accommodate structures like ideals, but modern standards often prioritize shared identity for consistency. Subrings play a central role in algebraic structures, facilitating the study of ring extensions, quotients, and homomorphisms; for example, the integers Z\mathbb{Z} are a subring of the rational numbers Q\mathbb{Q}, which in turn is a subring of the real numbers R\mathbb{R}. They also relate closely to ideals, which are subrings that absorb multiplication by elements of the parent ring, enabling constructions like quotient rings essential in fields such as number theory and algebraic geometry.

Definition and Variations

Formal Definition

In , a subring of a ring is formally defined as follows. Let (R,+,⋅)(R, +, \cdot) be a ring, where ++ denotes and ⋅\cdot denotes . A subset S⊆RS \subseteq R is a subring of RR, denoted (S,+∣S,⋅∣S)(S, +|_S, \cdot|_S), if SS is closed under the operations of RR restricted to SS (i.e., for all a,b∈Sa, b \in S, a+b∈Sa + b \in S and a⋅b∈Sa \cdot b \in S), SS contains the additive identity 0R0_R of RR, and SS is closed under additive inverses (i.e., for all s∈Ss \in S, −s∈S-s \in S). To verify that a SS of a ring RR is a subring, it suffices to confirm three conditions: closure under and as defined above, inclusion of the $0_R \in S,andclosureunderadditiveinversesforeveryelementin, and closure under additive inverses for every element in S.Equivalently,itsufficestocheckthat. Equivalently, it suffices to check that Scontainscontains0_R,isclosedunder[subtraction](/page/Subtraction),andclosedunder[multiplication](/page/Multiplication).[](https://math.libretexts.org/Bookshelves/AbstractandGeometricAlgebra/AnInquiry−BasedApproachtoAbstractAlgebra(Ernst)/08, is closed under [subtraction](/page/Subtraction), and closed under [multiplication](/page/Multiplication).[](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/08%3A_An_Introduction_to_Rings/8.01%3A_Definitions_and_Examples) These criteria ensure that S$ forms a ring under the induced operations without requiring a multiplicative identity, though variations incorporating unity are considered separately. The concept of a subring was introduced by in in his work on ideals. It was further developed in the early 20th-century axiomatic pioneered by through her axiomatic approach to ideals and rings in the 1920s, with formalization appearing in foundational texts by .

Unital and Non-Unital Subrings

In , the definition of a subring can vary depending on whether the multiplicative identity of the parent ring is required to be included. A unital subring SS of a ring RR with identity 1R1_R is a that forms a ring under the induced operations and contains 1R1_R, satisfying s⋅1R=1R⋅s=ss \cdot 1_R = 1_R \cdot s = s for all s∈Ss \in S. This ensures SS shares the same unit as RR, preserving the unital structure. Similarly, in texts emphasizing rings with identity, subrings are defined to include this element explicitly. In contrast, a non-unital subring (sometimes termed a rng subring, reflecting the absence of a required identity) omits this condition, requiring only that SS be closed under addition and multiplication, contain the 0R0_R, and be closed under additive inverses, forming an abelian under addition without necessarily including 1R1_R. This variation aligns with definitions of rings (or rngs) that do not mandate a multiplicative identity. The choice of definition has significant implications for ring structures and mappings. In unital rings, where homomorphisms are required to map 1R1_R to 1S1_S, unital subrings maintain compatibility with these maps, as the image of a unital ring under such a homomorphism remains unital. Non-unital subrings, however, allow for a wider array of subsets to qualify, such as ideals that lack the identity, but may not preserve homomorphism properties in unital contexts. For instance, the set 2Z2\mathbb{Z} of even integers is a non-unital subring of Z\mathbb{Z} under standard addition and multiplication, as it is closed under these operations and forms an additive subgroup, but it excludes 11 and thus fails the unital criterion. Contemporary algebra literature predominantly favors the unital subring convention, particularly in , to align with the standard assumption that rings possess a multiplicative identity and homomorphisms preserve it; seminal texts like Atiyah and Macdonald's Introduction to Commutative Algebra exemplify this approach. Earlier works, such as Herstein's Topics in Algebra, reflect a more permissive stance consistent with non-unital rings, though even there, unital cases are often highlighted when identities exist. This shift underscores the evolution toward unital structures in modern research for consistency in areas like and module theory.

Examples

Elementary Examples

The Z\mathbb{Z} is a subring of the field of rational numbers Q\mathbb{Q}. and in Q\mathbb{Q} restrict to those in Z\mathbb{Z}, as the sum and product of any two s are s. The 00 and multiplicative identity 11 of Q\mathbb{Q} both belong to Z\mathbb{Z}, and for every n∈Zn \in \mathbb{Z}, its −n-n is also in Z\mathbb{Z}. The set of even integers 2Z={2n∣n∈Z}2\mathbb{Z} = \{ 2n \mid n \in \mathbb{Z} \} forms a non-unital subring of Z\mathbb{Z}. It is closed under , since the sum of two even integers is even, and under , as the product of two even integers is even. The zero element 00 is included, and additive inverses exist, with −(2n)=2(−n)-(2n) = 2(-n) even for each 2n∈2Z2n \in 2\mathbb{Z}, but it lacks the multiplicative identity 11 of Z\mathbb{Z}. The set of 2×22 \times 2 upper triangular matrices over the real numbers R\mathbb{R}, consisting of matrices of the form (ab0c)\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} with a,b,c∈Ra, b, c \in \mathbb{R}, is a unital subring of the matrix ring M2(R)M_2(\mathbb{R}). Closure under addition holds because the sum of two such matrices has zeros below the diagonal. For multiplication, the product (a1b10c1)(a2b20c2)=(a1a2a1b2+b1c20c1c2)\begin{pmatrix} a_1 & b_1 \\ 0 & c_1 \end{pmatrix} \begin{pmatrix} a_2 & b_2 \\ 0 & c_2 \end{pmatrix} = \begin{pmatrix} a_1 a_2 & a_1 b_2 + b_1 c_2 \\ 0 & c_1 c_2 \end{pmatrix} is also upper triangular. The zero matrix and identity matrix (1001)\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} are upper triangular, and additive inverses preserve the form. In the polynomial ring R\mathbb{R}, the set of constant polynomials R0={f(x)=a∣a∈R}\mathbb{R}^0 = \{ f(x) = a \mid a \in \mathbb{R} \} forms a unital subring isomorphic to R\mathbb{R}. Addition of constants yields a constant, as does their multiplication, which is just the product in R\mathbb{R}. The zero polynomial and constant polynomial 11 serve as the identities, and the additive inverse of a constant aa is −a-a, also constant.

Non-Trivial Examples

In non-commutative ring theory, the real numbers R\mathbb{R} form a subring of the quaternion algebra H\mathbb{H}, where H\mathbb{H} consists of elements a+bi+cj+dka + bi + cj + dk with a,b,c,d∈Ra, b, c, d \in \mathbb{R} and the standard quaternion multiplication rules ensuring closure under addition and multiplication for scalar elements from R\mathbb{R}. This embedding highlights how commutative subrings can reside within non-commutative structures while preserving ring operations. In the polynomial ring Z\mathbb{Z}, the principal ideal (x)(x) generated by xx comprises all polynomials with integer coefficients and zero constant term, forming a non-unital subring closed under addition and multiplication since the product of any two such polynomials again has no constant term. This example illustrates how ideals in integral domains often yield proper subrings without the multiplicative identity of the parent ring. The ring of all real-valued functions on the interval [0,1][0,1], denoted R[0,1]\mathbb{R}^{[0,1]}, equipped with pointwise addition and multiplication, contains the continuous functions C[0,1]C[0,1] as a unital subring, as sums and products of continuous functions remain continuous. This infinite-dimensional case demonstrates subrings arising from topological constraints within larger function rings. In group ring constructions, the integral group ring ZG\mathbb{Z}G for a group GG is generated as a subring by the group elements {g∣g∈G}\{g \mid g \in G\} within the larger rational group algebra QG\mathbb{Q}G, where formal sums ∑ngg\sum n_g g with ng∈Zn_g \in \mathbb{Z} are closed under the induced operations. A to subring criteria is the set Q+\mathbb{Q}^+ of positive rational numbers within Q\mathbb{Q}, which fails to be a subring because it is not closed under additive inverses, as the negative of any positive rational lies outside Q+\mathbb{Q}^+.

Generation of Subrings

Subring Generated by a Set

In , for a ring RR and a subset S⊆RS \subseteq R, the subring generated by SS, denoted ⟨S⟩\langle S \rangle, is defined as the intersection of all subrings of RR that contain SS. This construction ensures that ⟨S⟩\langle S \rangle is itself a subring, as the intersection of subrings is a subring, and it contains SS by definition. Depending on the convention for subrings (as discussed in the article introduction), the generated subring may or may not be required to contain the multiplicative identity 1R1_R if [R](/page/R)[R](/page/R) is unital. In conventions where subrings share the identity, ⟨S⟩\langle S \rangle includes 1R1_R; otherwise, it may not. In a unital ring RR, under the unital subring convention, ⟨S⟩\langle S \rangle consists of all linear combinations of finite products of elements from SS (including the empty as 1). For commutative unital rings, this is the evaluation of polynomials in SS with coefficients. For non-unital rings or non-unital conventions, the construction excludes the identity and forms the smallest subring containing SS without it. An algorithmic perspective on generating ⟨S⟩\langle S \rangle involves starting with the set SS and iteratively adjoining additive inverses (i.e., −s-s for each s∈Ss \in S), sums of existing elements, and products of existing elements until closure under these operations is achieved; this process yields the desired subring in finitely many steps for any finite SS. The subring ⟨S⟩\langle S \rangle is unique, as it is the minimal subring containing SS with respect to inclusion, guaranteed by the intersection property.

Properties of Generated Subrings

The subring generated by a set SS in a ring RR is well-defined as the smallest subring containing SS, because the arbitrary of all subrings of RR that contain SS is itself a subring. This intersection property ensures that the generated subring exists and is unique. Unlike intersections, the union of subrings is not necessarily a subring. For example, in the ring Z\mathbb{Z} of integers, both 2Z2\mathbb{Z} (even integers) and 3Z3\mathbb{Z} (multiples of 3) are subrings, but their union contains 22 and 33 yet not 2+3=52 + 3 = 5, violating closure under . Even if the generating set SS is finite, the subring ⟨S⟩\langle S \rangle generated by SS may be infinite. For instance, in Z\mathbb{Z}, the subring generated by the singleton set S={2}S = \{2\} is 2Z2\mathbb{Z}, which consists of all even integers and is infinite. The subring generated by a set SS differs from the two-sided ideal generated by SS, which is the smallest two-sided containing SS and consists of all finite sums of elements of the form rstr s t where r,t∈Rr, t \in R and s∈Ss \in S. The generated ideal properly contains the generated subring in general, as it incorporates multiplications by arbitrary ring elements from both sides, whereas the subring only involves operations within the generated structure itself. In a RR with unity, the subring generated by a set S⊆RS \subseteq R consists precisely of all expressions in the elements of SS with coefficients, i.e., finite sums ∑ni∏j=1kisi,j\sum n_i \prod_{j=1}^{k_i} s_{i,j} where ni∈Zn_i \in \mathbb{Z}, ki≥0k_i \geq 0, and si,j∈Ss_{i,j} \in S (with the being 1).

Subrings in Ring Extensions

Adjoining Elements to Subrings

In , adjoining an element t∉St \notin S to a subring S⊆RS \subseteq R of a larger ring RR constructs the extension SS, the smallest subring of RR containing both SS and tt. This subring consists of all finite sums ∑i=0naiti\sum_{i=0}^n a_i t^i where ai∈Sa_i \in S and nn is finite, assuming tt satisfies some relation over SS; such elements form a basis for SS as an SS-module when tt is algebraic over SS. This process extends SS while preserving the ring structure, and SS coincides with the subring generated by S∪{t}S \cup \{ t \}. The construction of SS satisfies a universal property: it is isomorphic to the quotient ring S/IS / I, where SS is the over SS in an indeterminate xx, and II is the kernel of the ϕ:S→R\phi: S \to R defined by ϕ(f(x))=f(t)\phi(f(x)) = f(t) for f(x)∈Sf(x) \in S. This quotient identifies polynomials that evaluate to the same element at tt, ensuring SS is the universal ring extension of SS by an element satisfying the relations imposed by II, typically the ideal generated by the minimal polynomial of tt over SS if it exists. Any from SS to another ring that sends tt to some element factors uniquely through SS. A classic example is adjoining 2\sqrt{2}
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