What does integrality mean?
Definitions for integrality
in·te·gral·i·ty
This dictionary definitions page includes all the possible meanings, example usage and translations of the word integrality.
Princeton's WordNet
entirety, entireness, integrality, totalitynoun
the state of being total and complete
"he read the article in its entirety"; "appalled by the totality of the destruction"
Wiktionary
integralitynoun
The condition of being integral
Wikipedia
integrality
In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj in A such that b n + a n − 1 b n − 1 + ⋯ + a 1 b + a 0 = 0. {\displaystyle b^{n}+a_{n-1}b^{n-1}+\cdots +a_{1}b+a_{0}=0.} That is to say, b is a root of a monic polynomial over A. The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A. If every element of B is integral over A, then we say that B is integral over A, or equivalently B is an integral extension of A. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., 2 {\displaystyle {\sqrt {2}}} or 1 + i {\displaystyle 1+i} ); in this context, the integral elements are usually called algebraic integers. The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory. In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.
ChatGPT
integrality
Integrality generally refers to the state or quality of being whole, complete or undivided, often used in the context of mathematical concepts and functions. It is also used in integer programming in operations research, where variables are required to be integers, and in algebraic number theory to describe algebraic integers.
Webster Dictionary
Integralitynoun
entireness
Etymology: [Cf. F. intgralit.]
Numerology
Chaldean Numerology
The numerical value of integrality in Chaldean Numerology is: 3
Pythagorean Numerology
The numerical value of integrality in Pythagorean Numerology is: 5
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Translations for integrality
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"integrality." Definitions.net. STANDS4 LLC, 2024. Web. 26 Apr. 2024. <https://www.definitions.net/definition/integrality>.
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