INTEGRAL DOMAIN Meaning and
Definition
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An integral domain is a concept originating in abstract algebra and number theory, primarily used to classify certain types of algebraic structures. It is a commutative ring with unity, where neither zero-divisors nor non-zero zero-divisors exist.
Formally, an integral domain is defined as a commutative ring that satisfies the following conditions:
1. Closure under addition and multiplication: For any two elements a and b in the ring, their sum (a + b) and product (ab) are also in the ring.
2. Existence of unity: There exists an element 1 such that 1a = a1 = a for every element a in the ring.
3. No zero-divisors: If a and b are nonzero elements of the ring, then their product ab is also nonzero.
4. No non-zero zero-divisors: If ab = 0, where a and b are elements of the ring, then either a = 0 or b = 0.
An integral domain is characterized by the absence of divisors of zero, meaning two nonzero elements cannot multiply to give zero. This property ensures that the product of any two elements is nonzero as long as both elements are nonzero. Consequently, integral domains provide a foundation for various topics in algebra, including factorization theory, polynomial rings, and field theory.
Common Misspellings for INTEGRAL DOMAIN
- untegral domain
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- intrgral domain
Etymology of INTEGRAL DOMAIN
The word "integral domain" is formed by the combination of the words "integral" and "domain", both of which have distinct origins.
The word "integral" originated from the Latin word "integralis", which means "whole" or "complete". It entered the English language in the 16th century, referring to something that is necessary to make a whole or complete entity. In mathematics, the term "integral" is used to describe the concept of integration or the process of finding the area under a curve.
The word "domain" derives from the Old French word "domaine", which meant "estate" or "property". Its origin can be traced back to the Latin word "dominium", meaning "right of ownership" or "property". In mathematics, a "domain" refers to a specific subset of numbers or values for which a function or equation is defined.
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