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In [[algebraic geometry]] and related areas of [[mathematics]], '''local analy ==Group theory== ...

{{Short description|Part of ring theory in mathematics}} ...incipal ideal theorem]], a theorem on the height of ideals in a Noetherian Ring}} ...

{{Short description|1969 result in deformation theory}} ...tion theory]] which implies that [[formal power series]] with coefficients in a [[field (mathematics)|field]] ''k'' are well-approximated by the [[algebr ...

==Theorems of which articles are primarily devoted to proving them== ==Articles devoted to theorems of which a (sketch of a) proof is given== ...

{{short description|One of several theorems linking the sizes of different ideal class groups}} {{For|reflection principles in set theory|Reflection principle}} ...

...ften now called ''[[elementary algebra]]''. The distinction is rarely made in more recent writings. Structure preserving maps called ''homomorphisms'' are vital in the study of algebraic objects. ...

{{Short description|Measure theory and probability theorem}} ...ath> It is used as a type of [[transfinite induction]] to prove many other theorems, such as [[Fubini's theorem]]. ...

...to certain properties of [[total order#Chains|chain]]s of [[prime ideal]]s in [[integral extension]]s. ...aham Seidenberg]]. These are known as the '''going-up''' and '''going-down theorems'''. ...

.... [[Duality (category theory)|Dually]], a '''cocomplete category''' is one in which all small [[colimit]]s exist. A '''bicomplete category''' is a catego ==Theorems== ...

In [[mathematics]], a '''representation theorem''' is a [[theorem]] that state * [[Representation theory]] studies properties of abstract groups via their [[group representation|re ...

{{short description|Theorem in class field theory on mappings induced by extending ideals}} {{about|the Hauptidealsatz of class field theory|the theorem about Noetherian rings| Krull's principal ideal theorem}} ...

{{short description|Theorem in homotopy theory}} ...'. The conjecture was made in the mid 1970s by [[Graeme Segal]] and proved in 1984 by [[Gunnar Carlsson]]. This statement is still commonly referred to a ...

{{short description|Result in number theory, concerning irreducible polynomials}} ...nomials remain irreducible. This theorem is a prominent theorem in number theory. ...

{{short description|Theorem in abstract algebra}} The '''Latimer–MacDuffee theorem''' is a [[theorem]] in [[abstract algebra]], a branch of [[mathematics]]. ...

...mma''' is a technical result in [[abstract algebra]] used in [[algebraic K-theory]]. It states that a [[matrix (mathematics)|matrix]] of the form ...url=https://archive.org/details/explicitbrauerin0000snai/page/164 }}</ref> In symbols, ...

In [[mathematics]], the '''Herbrand–Ribet theorem''' is a result on the [[clas ...e ''n'', 0 < ''n'' < ''p'' − 1. The Herbrand–Ribet theorem specifies what, in particular, it means when ''p'' divides such an ''B_n''. ...

{{short description|Theorem in commutative algebra}} ...ng theory)|height]] of a [[principal ideal]] in a commutative [[Noetherian ring]]. The theorem is sometimes referred to by its German name, ''Krulls Haupt ...

...e [[automorphism]]s of [[simple ring]]s. It is a fundamental result in the theory of [[central simple algebra]]s. ...rie der assoziativen Zahlensysteme'' ([[German language|German]]: ''On the theory of associative number systems'') and later rediscovered by [[Emmy Noether]] ...

...al results factored out of proofs). See also [[list of axioms]], [[list of theorems]] and [[list of conjectures]]. {{Main category|Lemmas in algebra}} ...

{{Short description|Ring in abstract algebra}} ...descending chain condition for ideals simultaneously generalizes [[finite ring]]s and rings that are [[dimension (vector space)|finite-dimensional]] [[vec ...