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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, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. 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.
Whether the snake lemma holds in the category of groups depends on the definition of cokernel. If f : A → B {\displaystyle f:A\to B} is a homomorphism of groups, the universal property of the cokernel is satisfied by the natural map B → B / N ( im f ) {\displaystyle B\to B/N(\operatorname {im} f)} , where N ( im f ) {\displaystyle N ...
In mathematics, the Yoneda lemma is a fundamental result in category theory. [1] It is an abstract result on functors of the type morphisms into a fixed object . It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms).
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".
Mattel has modeled Barbie dolls after trailblazing activists, athletes, and celebrities. Barbie's Inspiring Women series includes Ida B. Wells, Rosa Parks, and Frida Kahlo.
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 ...
Dolls have been unveiled in honour of Women’s History Month and Barbie’s birthday