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The integral closure of a ring R in an extension ring S is the set of elements in S that are roots of monic polynomials with coefficients in R. It provides a way to extend the notion of 'closeness' or 'completeness' of a ring, similar to how real numbers complete the rational numbers.
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Ring theory is a branch of abstract algebra that studies rings, which are algebraic structures consisting of a set equipped with two binary operations that generalize the arithmetic of integers. It is fundamental in understanding structures such as fields, modules, and algebras, and has applications in number theory, geometry, and physics.
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A monic polynomial is a single-variable polynomial where the leading coefficient, the coefficient of the term with the highest degree, is equal to one. This property simplifies many algebraic operations and is particularly useful in polynomial division and factorization problems.
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An algebraic element over a field is an element that is a root of a non-zero polynomial with coefficients from that field, highlighting its significance in the study of field extensions. These elements contrast with transcendental elements, which do not satisfy any such polynomial equation, thus illustrating the boundary between algebraic and transcendental numbers or functions.
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A Noetherian ring is a ring in which every ascending chain of ideals terminates, ensuring that every ideal is finitely generated. This property is crucial in algebraic geometry and commutative algebra as it guarantees the ring has a well-behaved and manageable structure for studying polynomial equations and their solutions.
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A valuation ring is a commutative ring with unity in which, for any element from its field of fractions, either the element or its inverse is contained within the ring. This property makes valuation rings integral to understanding valuations in algebraic number theory and algebraic geometry, providing a local perspective on the divisibility of elements in a field.
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A Dedekind domain is an integral domain in which every nonzero proper ideal factors uniquely into a product of prime ideals, making it a generalization of the ring of integers. It plays a critical role in algebraic number theory and algebraic geometry due to its ideal-theoretic properties and connections to Noetherian rings and divisors.
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Normalization is a process in database design that organizes data to reduce redundancy and improve data integrity by dividing large tables into smaller, related tables. It involves applying a series of rules or normal forms to ensure that the database is efficient, consistent, and scalable.
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Algebraic closure of a field is the smallest field extension in which every polynomial equation with coefficients from the original field has a root. It is unique up to isomorphism and ensures that any polynomial can be completely factored into linear factors within this extended field.
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Krull dimension is a measure of the 'size' of a ring in terms of its prime ideal structure, reflecting the number of strictly increasing chains of prime ideals. It serves as a fundamental invariant in commutative algebra and algebraic geometry, linking algebraic properties of rings to geometric properties of varieties.
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Algebraic integers are a generalization of ordinary integers, defined as roots of monic polynomials with integer coefficients. They play a crucial role in number theory and algebraic geometry, particularly in the study of number fields and ring theory.
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An integral extension is a type of ring extension where elements of the larger ring satisfy a monic polynomial with coefficients in the smaller ring, making it a generalization of algebraic extensions in field theory. This concept is crucial in commutative algebra and algebraic geometry as it helps in understanding the structure and properties of rings and their spectra.
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An integrally closed domain is a commutative ring in which every element that is a root of a monic polynomial with coefficients from the ring is already an element of the ring itself. This property ensures that the ring is as 'complete' as possible with respect to the integral closure, meaning it contains all elements that should belong to it based on polynomial equations with coefficients in the ring.
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