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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/Local_ring
In the case of commutative rings, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal
https://en.wikipedia.org/wiki/Embedding
This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not
https://en.wikipedia.org/wiki/Skew-field
If one allows only rational instead of real coefficients in the constructions of the quaternions, one obtains another division ring
https://en.wikipedia.org/wiki/Measurable_function
If f : X → Y {\displaystyle f:X\to Y} is a measurable function, one writes f : ( X , Σ ) → ( Y , T )
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/List_of_representation_theory_topics
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https://en.wikipedia.org/wiki/Canonical_isomorphism
An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as is the case for solutions of a universal property), or if the isomorphism is much more natural (in some sense) than other isomorphisms
https://en.wikipedia.org/wiki/Ideal_(order_theory)
Now if any element n in M is such that n ∨ b is in F, one finds that (m ∨ n) ∨ b and (m ∨ n) ∨ a are both in F
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/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/Discrete_space
}"/>
A metric space ( E , d ) {\displaystyle (E,d)} is said to be uniformly discrete if there exists a packing radius 0"}}'>0}"> r > 0 {\displaystyle r>0} 0}"/> such that, for any x , y ∈ E , {\displaystyle x,y\in E,} one has either x = y {\displaystyle x=y} or r
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/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/Commutator_(ring_theory)
In linear algebra, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis
https://en.wikipedia.org/wiki/Unmixed_ideal
finitely generated) R-module M ≠ 0 {\displaystyle M\neq 0} is a Cohen–Macaulay module if d e p t h ( M ) = d i m ( M ) {\displaystyle \mathrm {depth} (M)=\mathrm {dim} (M)} (in general we have: d e p t h ( M ) ≤ d i m ( M ) {\displaystyle \mathrm {depth} (M)\leq \mathrm {dim} (M)} , see Auslander–Buchsbaum
https://en.wikipedia.org/wiki/Complex_analysis
Being able to move a given contour to a more suitable one often leads to major simplifications in proofs
https://en.wikipedia.org/wiki/Moment_generating_function
If the expectation does not exist in an open neighborhood of 0, we say that the moment generating function does not exist
https://en.wikipedia.org/wiki/Noncommutative_ring
It is common for noncommutative ring theorists to enforce a condition on one of these types of ideals while not requiring it to hold for the opposite side
https://en.wikipedia.org/wiki/Interval_(mathematics)
[1] If the infimum does not exist and the interval is not empty, one says often that the corresponding endpoint is negative infinity, written − ∞
https://en.wikipedia.org/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem
This is where the Lehmann–Scheffé theorem comes in, if T {\displaystyle T} is also complete, that is, for any real-valued measurable function φ {\displaystyle \varphi } it holds
E θ [ φ ( T ( X ) ) ] = 0 ⟹ φ ( T ( X ) ) = 0 P θ -a
https://en.wikipedia.org/wiki/Slavery_in_ancient_Egypt
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https://en.wikipedia.org/wiki/Ideal_(ring_theory)
If the ring is commutative, the definitions of left, right, and two-sided ideal coincide, and one talks simply of an ideal
https://en.wikipedia.org/wiki/White_noise
[3] In particular, if each sample has a normal distribution with zero mean, the signal is said to be additive white Gaussian noise
https://en.wikipedia.org/wiki/Semigroup
In mathematical analysis, the term also appears in the theory of one-parameter operator semigroups: see C0-semigroup
https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity
The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model
https://en.wikipedia.org/wiki/Projective_varieties
A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial