Examples of 'integral domain' in a sentence
Meaning of "integral domain"
integral domain: In mathematics, a type of algebraic structure that satisfies certain properties related to multiplication and addition
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- Any nonzero commutative ring in which the product of nonzero elements is nonzero.
How to use "integral domain" in a sentence
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integral domain
An inductive limit of integral domains is an integral domain.
The number ω is a unit in that integral domain.
An integral domain can be embedded in its quotient field.
The zero ring is not an integral domain.
Prove that in an integral domain every prime element is an irreducible.
This is the noncommutative generalization of integral domain.
A local ring that is an integral domain is called a local domain.
A commutative ring is a prime ring if and only if it is an integral domain.
Any matrix ring over an integral domain is a prime ring.
An integral domain is a ring.
Every field is an integral domain.
Every integral domain can carry at least the trivial absolute value.
It is thus an integral domain.
The integers have additional properties which make it an integral domain.
A principal ideal domain is an integral domain in which every ideal is principal.
See also
An integral domain is a commutative and unitary ring which has no proper zero divisors.
The same construction can be generalized to the field of fractions of any integral domain.
The reflector is the functor which sends each integral domain to its field of fractions.
An integral domain is a field if and only if its Krull dimension is zero.
A Prüfer domain is a semihereditary integral domain.
A Jaffard ring that is also an integral domain is called a Jaffard domain.
The analogous fact is false if A is not an integral domain.
If R is an integral domain it is injective.
The field F is the field of fractions of the integral domain OF.
The characteristic of an integral domain is either 0 or a prime number.
The ring of p-adic integers is an integral domain.
An integral domain or integral ring is a nontrivial ring without zero-divisors.
The entire functions on the complex plane form an integral domain in fact a Prüfer domain.
An integral domain in which Bézout 's identity holds is called a Bézout domain.
The ring Z is an integral domain.
Equivalently, an integral domain is a field if and only if its Krull dimension is 0.
If k is a field, then the polynomial ring k is an integral domain.
In particular, a commutative ring is an integral domain if and only if is a prime ideal.
On the other hand, not every reduced ring is an integral domain.
Theorem, Every finite integral domain is a field.
A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.
Conversely, every Artinian integral domain is a field.
An integral domain is a ring that is ( isomorphic to ) a subring of a field.
The results obtained are also valid if k is an integral domain of characteristic p > 2.
An integral domain is a commutative ring in which the zero ideal { 0 } is a prime ideal.
A principal ideal domain ( PID ) is an integral domain in which every ideal is principal.
Therefore, the ring of integers of F is an integral domain.
Every Frobenius algebra is self-injective, but no integral domain that is not a field is self-injective.
Indeed, one has the following characterization, let A be an integral domain.
Characteristic and homomorphisms ==The characteristic of an integral domain is either 0 or a prime number.
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