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https://en.wikipedia.org/wiki/Module_homomorphism
Explicitly, if M and N are left modules over a ring R, then a function f : M → N {\displaystyle f:M\to N} is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R,
f ( x + y ) = f ( x ) + f ( y ) , {\displaystyle f(x+y)=f(x)+f(y),} f ( r x ) = r f ( x )
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/Ideal_(ring_theory)
An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group
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/Semigroup
If the semigroup operation is commutative, then the semigroup is called a commutative semigroup or (less often than in the analogous case of groups) it may be called an abelian semigroup
https://en.wikipedia.org/wiki/Group_(mathematics)
In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory
https://en.wikipedia.org/wiki/Operator_algebra
From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator
https://en.wikipedia.org/wiki/Glossary_of_commutative_algebra
An ideal in a polynomial ring over a field is called absolutely prime if its extension remains prime for every extension of the field
https://en.wikipedia.org/wiki/Financial_economics
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https://en.wikipedia.org/wiki/List_of_abstract_algebra_topics
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https://en.wikipedia.org/wiki/Shing-Tung_Yau
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https://en.wikipedia.org/wiki/Double_centralizer_theorem
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https://en.wikipedia.org/wiki/Outline_of_category_theory
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https://en.wikipedia.org/wiki/Up_to
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https://en.wikipedia.org/wiki/Ideal_norm
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https://en.wikipedia.org/wiki/Conjugate_diameters
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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/Dobinski's_formula
[3] Recall that if x {\displaystyle x} and n {\displaystyle n} are nonnegative integers then the number of one-to-one functions that map a size- n {\displaystyle n} set into a size- x {\displaystyle x} set is the falling factorial
( x ) n = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) {\displaystyle (x)_{n}=x(x-1)(x-2
https://en.wikipedia.org/wiki/Gentoo_Linux
-- Weekly minimal install builds are available on Gentoo's site above the old 2008 releases -->"},"latest release date":{"wt":
https://en.wikipedia.org/wiki/Shaolin_temple
-- or | elevation_ft = -->"},"elevation_footnotes":{"wt":""},"designated":{"wt":""},"added":{"wt":""},"refnum":{"wt":""},"footnotes":{"wt":"{{Infobox UNESCO World Heritage Site\n| part_of = Historic Monuments of Dengfeng in \"The Centre of Heaven and Earth\"\n| child = yes\n| criteria = Cultural: (iv)\n| ID = 1305-005\
https://en.wikipedia.org/wiki/Artificial_intelligence
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https://en.wikipedia.org/wiki/Krull's_theorem
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https://en.wikipedia.org/wiki/List_of_differential_geometry_topics
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https://en.wikipedia.org/wiki/Bali_myna
[1]
In 2018, there were three locations on Bali where the birds existed in the wild: the West Bali National Park; Bali's small island of Nusa Penida and Begawan Foundation's breeding and release site at Melinggih Kelod, Payangan
https://en.wikipedia.org/wiki/Lithium%E2%80%93sulfur_battery
These constraints are interdependent: increasing sulfur loading or lowering E/S improves projected energy density, but can also increase polarization and lower reversible capacity if ion/electron transport and interfacial stability are not maintained
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/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/Diffusion_equation
The resulting diffusion algorithm can be written as an image convolution with a varying kernel (stencil) of size 3 × 3 in 2D and 3 × 3 × 3 in 3D
https://en.wikipedia.org/wiki/Android_(operating_system)
com/ndk/guides/abis |website=Android Developers |access-date=October 12, 2025}}</ref>)</small> -->"},"kernel type":{"wt":"[[Monolithic kernel|Monolithic]] ([[Linux kernel]])"},"userland":{"wt":"[[Bionic (software)|Bionic libc]], [[KornShell|mksh]] shell, [[Toybox]] as core utilities