Examples of 'principal ideal' in a sentence
Meaning of "principal ideal"
principal ideal ~ refers to a fundamental or primary concept, value, or belief that serves as a guiding principle or foundation for behavior or decision-making
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- An ideal I in an algebraic object R (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ R; the smallest ideal that contains a.
How to use "principal ideal" in a sentence
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principal ideal
All principal ideal domains are integrally closed.
In this case it is in fact a principal ideal domain.
Every principal ideal ring is factorial.
Show that it is a principal ideal ring.
A nonzero principal ideal is prime if and only if it is generated by a prime element.
An ideal generated by one element is called a principal ideal.
Modules over a principal ideal domain.
A principal ideal domain is an integral domain in which every ideal is principal.
Module over a principal ideal domain.
A principal ideal ring is a ring such that every ideal is principal.
Such an ideal is called a principal ideal.
Krull principal ideal theorem.
Finitely generated modules over a principal ideal domain.
Principal ideal domain.
Finitely generated module over a principal ideal domain.
See also
Z is a principal ideal domain.
Unique factorization domains and principal ideal domains.
This includes principal ideal domains and Euclidean domains.
Similar statements hold for any principal ideal domain.
All principal ideal domains and therefore all discrete valuation rings are Dedekind domains.
An ideal in a ring is termed a principal ideal if there exists an in such that.
Fundamental theorem of finitely generated modules over a principal ideal domain.
They asked me to prove that every principal ideal domain is a unique factorization domain.
A ring in which every ideal is principal is called principal, or a principal ideal ring.
An elementary proof that not all principal ideal domains are Euclidean domains.
R is a principal ideal domain with a unique non-zero prime ideal.
Prove that any nonzero prime ideal of a principal ideal domain is maximal.
Every principal ideal domain is an UFD.
The imaginary rings of quadratic integers that are principal ideal rings have been completely determined.
Every principal ideal domain is Noetherian.
All Euclidean domains and all fields are principal ideal domains.
Both above examples are principal ideal rings and also Euclidean domains for the norm.
Structure theorem for finitely-generated modules over a principal ideal domain.
As the Gaussian integers form a principal ideal domain they form also a unique factorization domain.
The Zariski-Samuel theorem determines the structure of a commutative principal ideal rings.
Let R be a principal ideal ring.
A principal ideal domain that is not a field has Krull dimension 1.
This implies that Z is a principal ideal domain.
Moreover, every principal ideal domain is a unique factorization domain.
The Zariski - Samuel theorem determines the structure of a commutative principal ideal rings.
In this case, the principal ideal generated by u is a prime ideal.
Finally, our results can be adapted to solve an associated problem, the Principal Ideal Problem.
Euclidean rings, principal ideal domains and factorial rings.
Then C is an ideal in R, and hence principal, since R is a principal ideal ring.
As for every principal ideal domain, is also a unique factorization domain.
A corollary is Krull 's principal ideal theorem.
A principal ideal domain ( PID ) is an integral domain in which every ideal is principal.
In particular, this is true if I is the principal ideal generated by a single element.
For a Bézout domain R, the following conditions are all equivalent, R is a principal ideal domain.
In particular, a commutative principal ideal domain which is not a field has global dimension one.
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