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...rnal|last1=Krull|first1=Wolfgang|title=Zur Theorie der zweiseitigen Ideale in nichtkommutativen Bereichen|journal=Mathematische Zeitschrift|volume=28|iss ...iplicative system (''i.e.'' <math>M</math> is closed under multiplication) in a [[Ring (mathematics)|ring]] <math>R</math>, and suppose ...

...ommutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, {{ISBN|0-387-94268-8}}, Exercise 4 ...nd <math>B \sub C</math> [[commutative algebra (structure)|commutative]] [[algebra over a ring|algebras over ''A'']]. If ''C'' is of finite type over ''A'' an ...

In mathematics, the '''Teichmüller–Tukey lemma''' (sometimes named just '''Tuk #For each <math>A\in \mathcal{F}</math>, every [[finite set|finite]] [[subset]] of <math>A</math ...

{{Short description|On a particular long exact sequence in the homology groups of certain chain complexes}} ...e [[homology group]]s of certain [[chain complex]]es. The result is valid in every [[abelian category]]. ...

{{Short description|Theorem in functional analysis}} ...ics, '''Auerbach's lemma''', named after [[Herman Auerbach]], is a theorem in [[functional analysis]] which asserts that a certain property of [[Euclidea ...

...zero. Historically, they are regarded as leading to the discovery of [[Lie algebra cohomology]].<ref>{{harvnb|Jacobson|1979|loc=p. 93}}</ref> ...ely, but there are similar statements pertaining to Lie algebra cohomology in arbitrary orders which are also attributed to Whitehead. ...

In [[mathematics]], '''Cartan's lemma''' refers to a number of results named a * In [[exterior algebra]]:<ref>*{{cite book | last = Sternberg | first = S. | year = 1983 | title = ...

...[[linear algebra]] used, for example, to show that any two [[Basis (linear algebra)|bases]] for a finite-[[Dimension (vector space)|dimensional]] [[vector spa ...ac Lane|year=1936|title=Some interpretations of abstract linear dependence in terms of projective geometry|journal=American Journal of Mathematics|volume ...

In [[mathematics]], more specifically [[group theory]], the '''three subgroups In what follows, the following notation will be employed: ...

...] had discovered it earlier, in 1953.<ref>{{citation|series=Lectures Notes in Mathematics|volume=1758|year=2001|title=Continuous Bounded Cohomology of Lo ...sentation of ''G'' and ''N'' a finite-dimensional representation of ''H''. In this case, the module ''S'' ⊗_''R'' ''N'' is called the [[induced ...

...} is [[finitely generated algebra|finitely generated]] as an [[associative algebra]] over another field {{math|''k''}}, then {{math|''K''}} is a [[finite fiel ...s algebraically closed that extension must be ''k''. Then for any <math>f \in \mathfrak{m}</math>, ...

...as ''g''(''x'')''h''(''x'') with ''h''(''x'') also having [[coefficient]]s in&nbsp;''F''.<ref name="dorrie">{{citation|title=100 Great Problems of Elemen .../09/abels-lemmas-on-irreducibility.html Fermat's Last Theorem blog: Abel's Lemmas on Irreducibility]. September 4, 2008. ...

...rkhoff–Witt theorem]] used in the construction of a [[universal enveloping algebra]]. As such, this generalizes the same process when done with the Lyndon wor ...ng process]] was described first, in 1934, by [[Philip Hall]] and explored in 1937 by [[Wilhelm Magnus]].<ref> ...

...lemma in [[convex geometry]] and [[algebraic geometry]]. It can be stated in several ways. ...f <math>A \cdot x = 0</math>. Then there exists a finite subset of vectors in <math>M</math>, such that every element of <math>M</math> is a linear combi ...

In [[number theory]], more specifically in [[p-adic analysis|''p''-adic analysis]], '''Krasner's lemma''' is a basic r ...{{overline|''K''}} be a [[separable closure]] of ''K''. Given an element α in {{overline|''K''}}, denote its [[Galois conjugate]]s by ''α''₂,& ...

In algebra, the '''local criterion for flatness''' gives conditions one can check to s |In the notations of 3., <math>M_0</math> is <math>A_0</math>-flat and the natu ...

{{Short description|Gröbner bases for non-commutative algebra}} ...ts defining relations. However, in contrast to [[Buchberger's algorithm]], in the non-commutative case, this algorithm may not terminate.<ref name=":0">R ...

In [[numerical linear algebra]], the [[conjugate gradient method]] is an [[iterative method]] for numeric The intent of this article is to document the important steps in these derivations. ...

...'''ring lemma''' gives a [[lower bound]] on the sizes of adjacent circles in a circle packing.{{r|s}} ...s in the ring tangent to each other. Then the minimum radius of any circle in the ring is at least the [[unit fraction]] ...

{{Short description|Theorem in abstract algebra}} ...newspaper=[[Time (magazine)|Time]] |access-date=December 14, 2009 }}</ref> In 2010, Ngô was awarded the [[Fields Medal]] for this proof. ...