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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/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/List_of_representation_theory_topics
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https://en.wikipedia.org/wiki/Elliott_H._Lieb
entropy principle]]<br>[[AKLT model|Affleck-Kennedy-Lieb-Tasaki model]]<br>[[Borell–Brascamp–Lieb inequality]]<br>[[Brascamp–Lieb inequality]]<br>[[Brezis–Lieb lemma]]<br>[[Strong subadditivity of quantum entropy#Carlen-Lieb Extension|Carlen–Lieb extension]]<br>[[Temperley–Lieb algebra]]<br>[[Dicke model]
https://en.wikipedia.org/wiki/List_of_topology_topics
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https://en.wikipedia.org/wiki/List_of_commutative_algebra_topics
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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/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/List_of_differential_geometry_topics
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https://en.wikipedia.org/wiki/Stephen_Smale
]]<br>[[Morse–Smale system]]<br>[[Morse–Smale diffeomorphism]]<br>[[Palais–Smale compactness condition]]<br>[[Blum–Shub–Smale machine]]<br>[[Smale–Williams attractor]]<br>[[Morse–Palais lemma]]<br>[[Regular homotopy]]<br>[[Sard's theorem]]<br>[[Sphere eversion]]<br>[[Structural stability]]<br>[[Whitehead
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
https://en.wikipedia.org/wiki/Ideal_(order_theory)
In this case, the following equivalent definition can be given: a subset I of a lattice ( P , ≤ ) {\displaystyle (P,\leq )} is an ideal if and only if it is a lower set that is closed under finite joins (suprema); that is, it is nonempty and for all x, y in I, the element x ∨ y {\displaystyle x\vee y} of P is also
https://en.wikipedia.org/wiki/Unmixed_ideal
But we can expand the definition for a more general Noetherian ring: If R {\displaystyle R} is a commutative Noetherian ring, then an R-module M is called Cohen–Macaulay module if M m {\displaystyle M_{\mathrm {m} }} is a Cohen–Macaulay module for all maximal ideals m ∈ S u p p ( M ) {\displaystyle \mathrm {m} \
https://en.wikipedia.org/wiki/Noether's_second_theorem
The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least action
https://en.wikipedia.org/wiki/Discrete_space
The discrete topology is the finest topology that can be given on a set
https://en.wikipedia.org/wiki/Commutator_(ring_theory)
[1]
This element is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg)
https://en.wikipedia.org/wiki/Embedding
[3] More explicitly, an injective continuous map f : X → Y {\displaystyle f:X\to Y} between topological spaces X {\displaystyle X} and Y {\displaystyle Y} is a topological embedding if f {\displaystyle f} yields a homeomorphism between X {\displaystyle X} and f ( X ) {\displaystyle f(X)} (where f ( X ) {\displaystyle
https://en.wikipedia.org/wiki/Projective_dimension
However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem)
https://en.wikipedia.org/wiki/Canonical_isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping
https://en.wikipedia.org/wiki/Injective_module
Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q
https://en.wikipedia.org/wiki/Monotonic
A function is termed monotonically increasing (also increasing or non-decreasing)[3] if for all x {\displaystyle x} and y {\displaystyle y} such that x ≤ y {\displaystyle x\leq y} one has f ( x ) ≤ f ( y ) {\displaystyle f\
https://en.wikipedia.org/wiki/Monotonic_function
A function is termed monotonically increasing (also increasing or non-decreasing)[3] if for all x {\displaystyle x} and y {\displaystyle y} such that x ≤ y {\displaystyle x\leq y} one has f ( x ) ≤ f ( y ) {\displaystyle f\
https://en.wikipedia.org/wiki/Noncommutative_ring
x n ⟩ / ( x i x j − q i j x j x i ) {\displaystyle \mathbb {C} \langle x_{1},\ldots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})} , called a quantum plane, where q i j ∈ C {\displaystyle q_{ij}\in \mathbb {C} }
Any Clifford algebra can be described explicitly using an algebra presentation: given an F {\
https://en.wikipedia.org/wiki/Moment_generating_function
As its name implies, the moment generating function can be used to compute a distribution’s moments: the n-th moment about 0 is the n-th derivative of the moment generating function, evaluated at 0
https://en.wikipedia.org/wiki/Non-associative_ring
That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative
https://en.wikipedia.org/wiki/Complex_analysis
}
A complex function is continuous if and only if its associated vector-valued function of two variables is also continuous
https://en.wikipedia.org/wiki/Truth_tables
[1] In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid