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Subring

In ring theory, a subring of a ring R is a nonempty subset S \subseteq R that is closed under the addition and multiplication operations inherited from R, contains the additive identity $0 of R, and is closed under additive inverses, thereby forming a ring itself under these operations.^[1]^[2] A standard test for verifying a subset S is a subring requires showing that S is nonempty, closed under subtraction (which ensures closure under addition and additive inverses), and closed under multiplication.^[1] Definitions of subrings vary slightly across mathematical literature, particularly regarding the multiplicative identity. In contexts emphasizing rings with unity (multiplicative identity $1), such as commutative algebra, subrings are often required to contain the same $1 as R, ensuring compatibility with units and ring homomorphisms.^[3]^[2] However, in more general treatments of ring theory, subrings need not include $1_R; for instance, the even integers $2\mathbb{Z} form a subring of the integers \mathbb{Z} despite lacking $1.^[1] This distinction arises from historical developments in ring theory, where early definitions sometimes omitted unity to accommodate structures like ideals, but modern standards often prioritize shared identity for consistency.^[3] Subrings play a central role in algebraic structures, facilitating the study of ring extensions, quotients, and homomorphisms; for example, the integers \mathbb{Z} are a subring of the rational numbers \mathbb{Q}, which in turn is a subring of the real numbers \mathbb{R}.^[1]^[2] 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.^[3]

Definition and Variations

Formal Definition

In ring theory, a subring of a ring is formally defined as follows. Let (R, +, \cdot) be a ring, where + denotes addition and \cdot denotes multiplication. A subset S \subseteq R is a subring of R, denoted (S, +|_S, \cdot|_S), if S is closed under the operations of R restricted to S (i.e., for all a, b \in S, a + b \in S and a \cdot b \in S), S contains the additive identity $0_R of R, and S is closed under additive inverses (i.e., for all s \in S, -s \in S).^[4] To verify that a subset S of a ring R is a subring, it suffices to confirm three conditions: closure under addition and multiplication as defined above, inclusion of the zero element $0_R \in S, and closure under additive inverses for every element in S. Equivalently, it suffices to check that Scontains0_R, is closed under [subtraction](/page/Subtraction), and closed under [multiplication](/page/Multiplication)./08%3A_An_Introduction_to_Rings/8.01%3A_Definitions_and_Examples) These criteria ensure that S$ forms a ring under the induced operations without requiring a multiplicative identity, though variations incorporating unity are considered separately.^[4] The concept of a subring was introduced by Richard Dedekind in 1871 in his work on ideals. It was further developed in the early 20th-century axiomatic ring theory pioneered by Emmy Noether through her axiomatic approach to ideals and rings in the 1920s, with formalization appearing in foundational texts by the 1930s.^[5]

Unital and Non-Unital Subrings

In ring theory, 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 S of a ring R with identity $1_R is a subset that forms a ring under the induced operations and contains $1_R, satisfying s \cdot 1_R = 1_R \cdot s = s for all s \in S. This ensures S shares the same unit as R, preserving the unital structure.^[6] 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 S be closed under addition and multiplication, contain the additive identity $0_R, and be closed under additive inverses, forming an abelian subgroup under addition without necessarily including $1_R. This variation aligns with definitions of rings (or rngs) that do not mandate a multiplicative identity.^[7] The choice of definition has significant implications for ring structures and mappings. In unital rings, where homomorphisms are required to map $1_R to $1_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 $2\mathbb{Z} of even integers is a non-unital subring of \mathbb{Z} under standard addition and multiplication, as it is closed under these operations and forms an additive subgroup, but it excludes $1 and thus fails the unital criterion. Contemporary algebra literature predominantly favors the unital subring convention, particularly in commutative algebra, 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 algebraic geometry and module theory.^[6]^[7]

Examples

Elementary Examples

The ring of integers \mathbb{Z} is a subring of the field of rational numbers \mathbb{Q}.^[3] Addition and multiplication in \mathbb{Q} restrict to those in \mathbb{Z}, as the sum and product of any two integers are integers.^[3] The zero element $0 and multiplicative identity $1 of \mathbb{Q} both belong to \mathbb{Z}, and for every integer n \in \mathbb{Z}, its additive inverse -n is also in \mathbb{Z}.^[3] The set of even integers $2\mathbb{Z} = \{ 2n \mid n \in \mathbb{Z} \} forms a non-unital subring of \mathbb{Z}.^[8] It is closed under addition, since the sum of two even integers is even, and under multiplication, as the product of two even integers is even.^[8] The zero element $0 is included, and additive inverses exist, with -(2n) = 2(-n) even for each $2n \in 2\mathbb{Z}, but it lacks the multiplicative identity $1 of \mathbb{Z}.^[8] The set of $2 \times 2 upper triangular matrices over the real numbers \mathbb{R}, consisting of matrices of the form \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} with a, b, c \in \mathbb{R}, is a unital subring of the matrix ring M_2(\mathbb{R}).^[9] Closure under addition holds because the sum of two such matrices has zeros below the diagonal.^[9] For multiplication, the product \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.^[9] The zero matrix and identity matrix \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} are upper triangular, and additive inverses preserve the form.^[9] In the polynomial ring \mathbb{R}, the set of constant polynomials \mathbb{R}^0 = \{ f(x) = a \mid a \in \mathbb{R} \} forms a unital subring isomorphic to \mathbb{R}.^[10] Addition of constants yields a constant, as does their multiplication, which is just the product in \mathbb{R}.^[10] The zero polynomial and constant polynomial $1 serve as the identities, and the additive inverse of a constant a is -a, also constant.^[10]

Non-Trivial Examples

In non-commutative ring theory, the real numbers \mathbb{R} form a subring of the quaternion algebra \mathbb{H}, where \mathbb{H} consists of elements a + bi + cj + dk with a, b, c, d \in \mathbb{R} and the standard quaternion multiplication rules ensuring closure under addition and multiplication for scalar elements from \mathbb{R}.^[11] This embedding highlights how commutative subrings can reside within non-commutative structures while preserving ring operations. In the polynomial ring \mathbb{Z}, the principal ideal (x) generated by x 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.^[12] 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], denoted \mathbb{R}^{[0,1]}, equipped with pointwise addition and multiplication, contains the continuous functions C[0,1] as a unital subring, as sums and products of continuous functions remain continuous.^[13] This infinite-dimensional case demonstrates subrings arising from topological constraints within larger function rings. In group ring constructions, the integral group ring \mathbb{Z}G for a group G is generated as a subring by the group elements \{g \mid g \in G\} within the larger rational group algebra \mathbb{Q}G, where formal sums \sum n_g g with n_g \in \mathbb{Z} are closed under the induced operations.^[14] A counterexample to subring criteria is the set \mathbb{Q}^+ of positive rational numbers within \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 \mathbb{Q}^+.^[15]

Generation of Subrings

Subring Generated by a Set

In ring theory, for a ring R and a subset S \subseteq R, the subring generated by S, denoted \langle S \rangle, is defined as the intersection of all subrings of R that contain S. This construction ensures that \langle S \rangle is itself a subring, as the intersection of subrings is a subring, and it contains S by definition.^[16] 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 $1_R if [R](/page/R) is unital. In conventions where subrings share the identity, \langle S \rangle includes $1_R; otherwise, it may not. In a unital ring R, under the unital subring convention, \langle S \rangle consists of all integer linear combinations of finite products of elements from S (including the empty product as 1). For commutative unital rings, this is the evaluation of polynomials in S with integer coefficients. For non-unital rings or non-unital conventions, the construction excludes the identity and forms the smallest subring containing S without it.^[17] An algorithmic perspective on generating \langle S \rangle involves starting with the set S and iteratively adjoining additive inverses (i.e., -s for each s \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 S. The subring \langle S \rangle is unique, as it is the minimal subring containing S with respect to inclusion, guaranteed by the intersection property.^[18]

Properties of Generated Subrings

The subring generated by a set S in a ring R is well-defined as the smallest subring containing S, because the arbitrary intersection of all subrings of R that contain S is itself a subring.^[13] 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 \mathbb{Z} of integers, both $2\mathbb{Z} (even integers) and $3\mathbb{Z} (multiples of 3) are subrings, but their union contains $2 and $3 yet not $2 + 3 = 5, violating closure under addition.^[19] Even if the generating set S is finite, the subring \langle S \rangle generated by S may be infinite. For instance, in \mathbb{Z}, the subring generated by the singleton set S = \{2\} is $2\mathbb{Z}, which consists of all even integers and is infinite.^[13] The subring generated by a set S differs from the two-sided ideal generated by S, which is the smallest two-sided ideal containing S and consists of all finite sums of elements of the form r s t where r, t \in R and s \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.^[20] In a commutative ring R with unity, the subring generated by a set S \subseteq R consists precisely of all polynomial expressions in the elements of S with integer coefficients, i.e., finite sums \sum n_i \prod_{j=1}^{k_i} s_{i,j} where n_i \in \mathbb{Z}, k_i \geq 0, and s_{i,j} \in S (with the empty product being 1).^[17]

Subrings in Ring Extensions

Adjoining Elements to Subrings

In ring theory, adjoining an element t \notin S to a subring S \subseteq R of a larger ring R constructs the extension S, the smallest subring of R containing both S and t.^[21] This subring consists of all finite sums \sum_{i=0}^n a_i t^i where a_i \in S and n is finite, assuming t satisfies some polynomial relation over S; such elements form a basis for S as an S-module when t is algebraic over S.^[22] This process extends S while preserving the ring structure, and S coincides with the subring generated by S \cup \{ t \}.^[21] The construction of S satisfies a universal property: it is isomorphic to the quotient ring S / I, where S is the polynomial ring over S in an indeterminate x, and I is the kernel of the evaluation homomorphism \phi: S \to R defined by \phi(f(x)) = f(t) for f(x) \in S. This quotient identifies polynomials that evaluate to the same element at t, ensuring S is the universal ring extension of S by an element satisfying the relations imposed by I, typically the ideal generated by the minimal polynomial of t over S if it exists.^[23] Any ring homomorphism from S to another ring that sends t to some element factors uniquely through S.^[21] A classic example is adjoining \sqrt{2} to the rationals \mathbb{Q} \subseteq \mathbb{R}, yielding \mathbb{Q}[\sqrt{2}] = \{ a + b \sqrt{2} \mid a, b \in \mathbb{Q} \}, which is isomorphic to \mathbb{Q} / (x^2 - 2) since \sqrt{2} satisfies the monic polynomial x^2 - 2 = 0.^[22] Here, \{1, \sqrt{2}\} forms a basis over \mathbb{Q}, and \mathbb{Q}[\sqrt{2}] is a degree-2 extension.^[23] This adjoining process is typically discussed under the assumption that S is commutative (often an integral domain), allowing the use of ordinary polynomial rings where powers of t commute with elements of S.^[21] In the non-commutative case, adjoining t to S instead involves constructing a free algebra S \langle t \rangle, the non-commutative analogue of the polynomial ring, generated by all finite non-commuting words in elements of S and t.^[24] If t is not algebraic over S (i.e., transcendental), then S is isomorphic to the polynomial ring S, and the embedding into R may not be "nice" in the sense of producing a finite-dimensional extension, as it requires infinitely many distinct powers of t to remain linearly independent over S.^[22]

Prime Subring and Characteristic

In a unital ring R, the prime subring, often denoted \mathbb{Z} \cdot 1 or Z(R), is the subring generated by the multiplicative identity element $1_R. This subring consists of all integer multiples of $1_R, that is, \{ n \cdot 1_R \mid n \in \mathbb{Z} \}, and it forms the smallest subring of R containing the identity.^[13]^[25] The characteristic of R, denoted \operatorname{char}(R), is defined as the smallest positive integer n such that n \cdot 1_R = 0_R if such an n exists, or $0 otherwise. The prime subring encodes this characteristic: it is isomorphic to \mathbb{Z} if \operatorname{char}(R) = 0, and to \mathbb{Z}/n\mathbb{Z} if \operatorname{char}(R) = n > 0. More precisely, there is a canonical ring isomorphism Z(R) \cong \mathbb{Z} / \operatorname{char}(R) \mathbb{Z}, where \mathbb{Z}/0\mathbb{Z} is understood as \mathbb{Z}. This isomorphism arises from the universal property of the integers as the initial ring, mapping k \in \mathbb{Z} to k \cdot 1_R in Z(R), with the kernel being the ideal generated by \operatorname{char}(R).^[13]^[26] Every unital ring R possesses a unique prime subring, as it is the intersection of all subrings containing $1_R or, equivalently, the subring generated solely by the identity. For example, in the field of real numbers \mathbb{R}, \operatorname{char}(\mathbb{R}) = 0 and the prime subring is \mathbb{Z} \cdot 1 \cong \mathbb{Z}. In contrast, for the field \mathbb{Z}/p\mathbb{Z} where p is prime, \operatorname{char}(\mathbb{Z}/p\mathbb{Z}) = p and the prime subring coincides with the entire ring.^[13]^[25]^[26]

Integral Extensions Involving Subrings

In ring theory, an integral extension occurs when S is a subring of R and every element r ∈ R satisfies a monic polynomial equation with coefficients in S, denoted as R being integral over S.^[27] This condition implies that R is an S-module, with each element r generating a finitely generated submodule S isomorphic to S/(f(x)) for some monic polynomial f ∈ S.^[28] From the subring perspective, such extensions preserve key structural properties, such as the ring operations being compatible, while emphasizing the algebraic dependence of R on S. A fundamental result in this context is the lying-over theorem, which states that if R is integral over S, then for every prime ideal P of S, there exists a prime ideal Q of R such that Q ∩ S = P.^[29] This theorem ensures that prime ideals "lift" across the extension, facilitating the study of Spec(R) over Spec(S) and underscoring the controlled behavior of ideals in integral settings.^[30] A classic example is the extension ℤ ⊆ ℤ[√d], where d is a square-free integer; here, √d satisfies the monic polynomial x² - d = 0 over ℤ, making the extension integral, though ℤ[√d] does not contain all algebraic integers of ℚ(√d) when d ≡ 1 (mod 4). For instance, when d = 5, the element (1 + √5)/2 lies in the full ring of integers but not in ℤ[√5], highlighting that ℤ[√d] is a proper integral subring.^[31] The integral closure of S in R, often called the normalization of S, is the largest subring of R consisting of elements integral over S, and it is itself integrally closed in its fraction field.^[32] This closure provides a canonical way to extend S to a "normal" ring within R, capturing all integral dependencies while remaining a subring.^[33]

References

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https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/08:_An_Introduction_to_Rings/8.01:_Definitions_and_Examples
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[PDF] Rings and Subrings - Princeton University
Definition 1.2. A subset S of a ring R is a subring of R if S is closed under the addition and multiplication operations of R, contains additive inverses, and ...
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[PDF] Standard definitions for rings - Keith Conrad
Then kernels of ring homomorphisms could be called subrings. The development of ring theory, particularly for commutative rings, has shown that this is a bad ...
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Ring Theory - MacTutor History of Mathematics
However, axioms for rings are not given by Weber and the axiomatic treatment of commutative rings was not developed until the 1920's in the work of Emmy Noether ...
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[PDF] Introduction to Commutative Algebra
A subset S of a ring A is a subring of A if S is closed under addition and multiplication and contains the identity element of A. The identity mapping of. S ...
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[PDF] Abstract Algebra
a divides b the greatest common divisor of a, b also the ideal generated by a, b the order of the set A, the order of the element x.
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[PDF] topics in algebra - Mathematics Area
Page 1. Page 2. i. n. herstein. University of Chicago. TOPICS IN. ALGEBRA. 2nd ... Definition of a Group. Some Examples of Groups. Some Preliminary Lemmas.
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[PDF] Chapter 3, Rings Definitions and examples. We now have several ...
2Z = { 2n | n ∈ Z} is a subring of Z, but the only subring of Z with identity is Z itself. The zero ring is a subring of every ring.
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[PDF] Abstract Algebra I - Lecture 6 - Michigan State University
Sep 19, 2015 · so it is also closed under multiplication. Upper triangular matrices also form a subring: The crucial part is the closure under ...
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[PDF] 22 Rings of Polynomials - UCI Mathematics
To confuse matters, the ring R is naturally isomorphic to the subring of constant polynomials. {f(x) = a : a ∈ R} ≤ R[x]. This subring is usually referred to ...
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[PDF] Ring Theory - The University of Memphis
We can also define the subring generated by a subset. More generally, if R is a subring of R. ′ and S ⊆ R. ′. , then R[S] is the smallest subring of R. ′.<|control11|><|separator|>
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[PDF] Structure of group rings and the group of units of integral ... - arXiv
Aug 26, 2020 · Of course ZG is a subring of the rational group algebra QG; and thus one can talk of (Q-) independent elements in ZG. Lemma 2.1 Let G be a ...
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Sample solution (sketch): This isn't closed under taking additive inverse, since the negative of a (strictly) positive rational number is (strictly) negative.
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[PDF] Graduate Algebra Notes Sean Sather-Wagstaff
If R is commutative, then so is the subring generated by X. If R has identity and 1R ∈ X, then the subring generated by X is a subring with identity. 1R. If ...
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None
Summary of each segment:
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[PDF] Math 371 Assignment 2
Now consider. 2Z and 3Z as ideals of Z. Their set-theoretic union contains 2 and 3 but not 2 + 3 = 5 since. 5 isn't a Z-multiple of either 2 or 3. 4. Let R be ...
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[PDF] Ideals and Subrings
Apr 8, 2018 · Likewise, a subring of a ring is a subset of the ring which is a ring in its own right, using the addition and multiplication it inherits from ...
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[PDF] ring theory - Northwestern University
The polynomial ring plays the same role for A-algebras that the free group plays for groups. Theorem. Let A and B be commutative rings, and let j : A → B be a ...
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[PDF] RES.18-012 (Spring 2022) Lecture 9: Building New Rings
9.3 Adjoining Elements to a Ring. A different way of creating new rings, which is quite important, is to adjoin elements. Definition 9.5. If R is a subring of ...
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[PDF] Algebra, Second Edition - CSE IITB
... Adjoining E le m e n ts ... Ring. 14.1 Modules...............................................................................................
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Feb 1, 2007 · 4.1 Adjoining one element to a ring. Let R be a ring (with the ... A ring A is said to be algebraic over a subring B if every element of A is.
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[PDF] Rings whose subrings have an identity - UCCS Faculty Sites
1Some authors define a subring S of a ring R with identity 1R to be unital if 1R ∈ S. In fact, it is commonplace. for many authors to consider only rings with ...
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Identification: We identify the prime subring of a ring of characteristic n with Zn if n > 0 and Z if n = 0. Thus using this identification, we have that ...
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[PDF] Integral Ring Extensions
We say that the ring B is integral over A if every element of B is integral over A. For any b ∈ B, there is the subring A[b] ⊂ B, the smallest subring of B ...
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[PDF] Integrality and valuation rings - UChicago Math
Integrality, similar to algebraic, requires monic polynomials. An element is integral over a ring if it satisfies a monic polynomial equation in the ring.
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[PDF] 3 Extensions of Rings and Valuations
Theorem 3.10 (Lying Over Theorem) Suppose A ⇢ B are domains, B is integral over A and P is a prime ideal of A. There is a prime ideal Q of B such that A ...
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Lying Over - Purdue Math
Lying over means a prime ideal lies over another, and the lying over property means for any prime ideal, there exists a prime ideal that lies over it.
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[PDF] Math 210B. Quadratic integer rings 1. Computing the integral ...
If d ≡ 2,3 mod 4 then OK = Z[. √ d], and if d ≡ 1 mod 4 then OK = Z[(1 +. √ d)/2]. Note that the case d ≡ 0 mod 4 cannot occur since d is square-free. Although ...
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[PDF] integral extensions, normal rings
(5.7) Definition: (a) Let RCS be an extension of rings. The intermedial ring R = { bES | b integral. over R} is called the integral closure of R in S. If R = R ...
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Section 10.37 (037B): Normal rings—The Stacks project
If S is a normal domain, then the integral closure of R in S is a normal domain. ... Since the set of almost integral elements form a subring (Lemma 10.37.