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More formally, proposition B is a corollary of proposition A, if B can be readily deduced from A or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, [4] which makes the theorem easier to use and apply, [5] even though its importance is generally considered to be secondary to that of ...
The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears. [ 4 ] Statement 3 : If M {\displaystyle M} is a finitely generated module over R {\displaystyle R} , J ( R ) {\displaystyle J(R)} is the Jacobson radical of R {\displaystyle R} , and J ( R ) M = M {\displaystyle J(R)M=M} , then M = 0 ...
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".
In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, ... Corollary [1] — If : is smooth ...
A specialized theorem that is mainly used to prove another theorem is called a lemma. A proven instance that forms part of a more general finding is termed a corollary. [98] Numerous technical terms used in mathematics are neologisms, such as polynomial and homeomorphism. [99]
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology .
Indiana may not have been one of the very best teams in college football, but a team that goes 11-1 in a power conference was always going to be a foregone conclusion to make the 12-team College ...
Another related result is the so-called counterpart of the Borel–Cantelli lemma. It is a counterpart of the Lemma in the sense that it gives a necessary and sufficient condition for the limsup to be 1 by replacing the independence assumption by the completely different assumption that ( A n ) {\displaystyle (A_{n})} is monotone increasing for ...