Examples of 'prime ideal' in a sentence
Meaning of "prime ideal"
prime ideal: In mathematics, particularly in the field of algebra, a prime ideal is a certain type of subset of a ring that has specific properties. Prime ideals play a significant role in abstract algebra and ring theory
Show more definitions
- Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I.
- In a commutative ring, a (two-sided) ideal I such that for arbitrary ring elements a and b, ab∈I⟹a∈I or b∈I.
How to use "prime ideal" in a sentence
Basic
Advanced
prime ideal
Show that every prime ideal in is maximal.
The radical of a primary ideal is a prime ideal.
Every nonzero prime ideal in it is maximal.
Show that the ideal is a prime ideal of.
A prime ideal is a radical ideal.
Every countable commutative ring has a prime ideal.
Every prime ideal is irreducible.
A prime element of a ring is an element that generates a prime ideal.
Further prime ideal theorems.
The central notion in commutative algebra is that of prime ideal.
Boolean prime ideal theorem.
A ring is prime if and only if its zero ideal is a prime ideal.
Every prime ideal is maximal.
This is equivalent to saying that the zero ideal is a prime ideal.
The preimage of a prime ideal under a ring homomorphism is a prime ideal.
See also
So the localization of a Bézout domain at a prime ideal is a valuation domain.
Prime ideal topology.
This article focuses on prime ideal theorems from order theory.
Prime ideal spectrum.
Height of a prime ideal.
Prime ideal of a valuation.
P is a prime ideal.
A positive integer n is a prime number if and only if nZ is a prime ideal in Z.
Prove that any nonzero prime ideal of a principal ideal domain is maximal.
Every proper ideal I in a ring has at least one minimal prime ideal above it.
Noncommutative rings in which every prime ideal is contained in a unique maximal ideal.
In this case, the principal ideal generated by u is a prime ideal.
A Jacobson ring is a ring such that every prime ideal is an intersection of maximal ideals.
The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras.
In particular, a commutative ring is an integral domain if and only if is a prime ideal.
Every quotient ring of R by a prime ideal has a zero Jacobson radical.
Every prime ideal is primary, but not conversely.
In a commutative ring with unity, every maximal ideal is a prime ideal.
Every prime ideal of A is principal.
R is a principal ideal domain with a unique non-zero prime ideal.
Thus, if a prime ideal is principal, it is equivalently generated by a prime element.
Equivalently, a set is a prime filter if and only if its complement is a prime ideal.
In a UFD, every nonzero prime ideal contains a prime element.
An integral domain is a commutative ring in which the zero ideal { 0 } is a prime ideal.
A is a Krull domain and every prime ideal of height 1 is principal.
The weak prime ideal theorem for Boolean algebras simply states,.
Localize about 2 and find a ring wherein the only prime ideal is principal.
Finally, prime ideal theorems do also exist for other ( not order-theoretical ) abstract algebras.
Gives many equivalent statements for the BPI, including prime ideal theorems for other algebraic structures.
Prime ideal theorems = = Recall that an order ideal is a ( non-empty ) directed lower set.
In particular, a prime ideal is primary.
It remains to show that $ P $ is a prime ideal.
For example, a field is zero-dimensional, since the only prime ideal is the zero ideal.
Actually, P is even a prime ideal.
You'll also be interested in:
Examples of using Ideal
Show more
Semipermeable tarp ideal for composting applications
An ideal solution to have an emergency backup plan
That house is ideal for your wife
Examples of using Prime
Show more
Prime minister will go through the roof
Effective safeguarding of prime movers and transmissions
The prime minister would not tell me
