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If an admissible rule can be expressed through a combination of the system's basic rules, it is called a derived or derivable rule
from the top result below
https://en.wikipedia.org/wiki/Inference_rule
If an argument with true premises follows a rule of inference then the conclusion cannot be false
https://en.wikipedia.org/wiki/Outline_of_finance
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https://en.wikipedia.org/wiki/Collusion
Because competition among sellers can provide consumers with low prices, conspiracy agreements increase the price consumers pay for the goods
https://en.wikipedia.org/wiki/Double_centralizer_theorem
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https://en.wikipedia.org/wiki/Up_to
For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation R that relates two lists if one can be obtained by reordering (permuting) the other
https://en.wikipedia.org/wiki/Conjugate_diameters
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https://en.wikipedia.org/wiki/Ideal_norm
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https://en.wikipedia.org/wiki/Wedderburn%E2%80%93Artin_theorem
If R is a finite-dimensional semisimple k-algebra, then each Di in the above statement is a finite-dimensional division algebra over k
https://en.wikipedia.org/wiki/Invariant_basis_number
Assuming the ultrafilter lemma (a strictly weaker form of the axiom of choice), this result is actually equivalent to the definition given here, and can be taken as an alternative definition
https://en.wikipedia.org/wiki/Theorem
[1]
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven
https://en.wikipedia.org/wiki/Saul_Kripke
|edition=Fall 2016 |language=en}}</ref> [[direct reference theory]], criticism of the [[Frege–Russell view]])<br />[[Admissible ordinal]]<br />[[Kripke structure]]<br />[[Rigid designator|Rigid]] vs
https://en.wikipedia.org/wiki/Grammatical_gender
In linguistics, a grammatical gender system is a specific form of a noun class system, where nouns are assigned to gender categories that are often not related to the real-world qualities of the entities denoted by those nouns
https://en.wikipedia.org/wiki/Markov_chain
Markov chains have many applications as statistical models of real-world processes
https://en.wikipedia.org/wiki/Tragedy_of_the_commons
He postulated that if a herder put more than his allotted number of cattle on the common, overgrazing could result
https://en.wikipedia.org/wiki/British_constitution
The House of Lords remains unelected but can be overruled
https://en.wikipedia.org/wiki/Nakayama's_lemma
Informally, Nakayama's lemma says that one can still regard a coherent sheaf as coming from a vector bundle in some sense
https://en.wikipedia.org/wiki/Outline_of_category_theory
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https://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem
that ideals in a Boolean algebra can be extended to prime ideals
https://en.wikipedia.org/wiki/Krull's_theorem
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https://en.wikipedia.org/wiki/Prime_ideal
This generalizes the following property of prime numbers, known as Euclid's lemma: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b
https://en.wikipedia.org/wiki/Stochastic_calculus
This field was created and started by the Japanese mathematician Kiyosi Itô during World War II
https://en.wikipedia.org/wiki/Basis_(linear_algebra)
: bases) if every element of V can be written in a unique way as a finite linear combination of elements of B
https://en.wikipedia.org/wiki/Faithful_module
For example, if R {\displaystyle R} is a commutative ring and I {\displaystyle I} is an ideal of R {\displaystyle R} , we can consider the quotient ring R / I {\displaystyle R/I} to be an R {\displaystyle R} -module
https://en.wikipedia.org/wiki/Integral_extension
over a subring A of B if b is a root of some monic polynomial over A
https://en.wikipedia.org/wiki/Measurable_function
} A function f : X → Y {\displaystyle f:X\to Y} is said to be measurable if for every E ∈ T {\displaystyle E\in \mathrm {T} } the pre-image of E {\displaystyle E} under f {\displaystyle f} is in Σ {\displaystyle \Sigma } ; that is, for all E ∈ T {\displaystyle E\in \mathrm {T} } f − 1 ( E ) := { x ∈ X ∣ f
https://en.wikipedia.org/wiki/Local_ring
The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper (principal) (left) ideals, where two ideals I1, I2 are called coprime if 1</sub> + ''I''<sub>2</sub>"}},"i":0}}]}'>R = I1 + I2
https://en.wikipedia.org/wiki/List_of_differential_geometry_topics
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https://en.wikipedia.org/wiki/Toll_(fee)
In Europe, the road toll goes back to the practice of the Germanic tribes, who charged fees to travellers if they wanted to cross over mountain passages
https://en.wikipedia.org/wiki/Consistent_estimator
If the sequence of estimates can be mathematically shown to converge in probability to the true value θ0, it is called a consistent estimator; otherwise the estimator is said to be inconsistent
https://en.wikipedia.org/wiki/Skew-field
In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring;[6] every division ring arises in this fashion from some simple module