A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmata, such as Bézout's lemma, Urysohn's lemma, Dehn's lemma, Euclid's lemma, Farkas ' lemma, Fatou's lemma, Gauss's lemma, Nakayama's lemma, Poincaré's lemma, Riesz's lemma, Schwarz's lemma, Itō's lemma and Zorn's lemma.

The Jacobson radical plays a prominent role in many ring and module theoretic results, such as Nakayama's lemma.

* Nakayama's lemma

Moreover, if R is a noetherian integral domain, then, by Nakayama's lemma, these conditions are equivalent to

Coherence of sheaves is working in the background of some results in commutative algebra, e. g. Nakayama's lemma, which in terms of sheaves says that if is a coherent sheaf, then the fiber if and only if there is a neighborhood of so that.

In mathematics, more specifically modern algebra and commutative algebra, Nakayama's lemma also known as the Krull – Azumaya theorem governs the interaction between the Jacobson radical of a ring ( typically a commutative ring ) and its finitely generated modules.

The following is Nakayama's lemma, as stated in:

The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears.

The following result manifests Nakayama's lemma in terms of generators

In this form, Nakayama's lemma takes on concrete geometrical significance.

Informally, Nakayama's lemma says that one can still regard a coherent sheaf as coming from a vector bundle in some sense.

Nakayama's lemma implies that a basis of the fibre F ( p ) lifts to a minimal set of generators of F < sub > p </ sub >.

The going up theorem is essentially a corollary of Nakayama's lemma.

To prove Nakayama's lemma from the Cayley – Hamilton theorem, assume that IM = M and take φ to be the identity on M. Then define a polynomial p ( x ) as above.

Nakayama's lemma makes precise one sense in which finitely generated modules over a commutative ring are like vector spaces over a field.

Instead of running over a large course as is the case in other countries, the course for the 4, 250 m ( about 2⅝ mile ) Nakayama Grand Jump follows a twisted path on the inside portion of Nakayama's racing ovals.

Followers of Tenrikyo believe that God, known by several names including Tenri-O-no-Mikoto, expressed divine will through Nakayama's role as the Shrine of God, and to a lesser extent the roles of the Honseki Izo Iburi and other leaders.

It is also true that Miki Nakayama's son, Shuji, sought and obtained approval and protection from the powerful Yoshida branch of Shinto at a relatively early stage in Tenrikyo's development ; he did this, however, contrary to his mother's wishes.

He was one of several instructors who demonstrated techniques in Nakayama's books on karate.

In 1985, two years before Nakayama's death, Abe was appointed as Director of Qualifications in the JKA.

Nakayama's songs were based on Japanese folk music called min ' yō, but also adopted Western musical style.

Now in the twilight of his career, injuries and age have taken a toll on Nakayama's skills but he still remains a favorite of the Jubilo faithful, as evidenced by the fact that he draws the loudest cheers by far from the home crowd at Yamaha Stadium when his name is announced during warm-ups or when he comes on as a substitute.

Nakayama's lemma also has several versions in homological algebra.

This is essentially the statement of Nakayama's lemma.

There is also a graded version of Nakayama's lemma.

The following consequence of Nakayama's lemma gives another way in which this is true:

Let be the discounted stock price given by, then by Ito's lemma we get the SDE:

Let F be a functor from the category of ringed spaces to the category of sets, and let G ⊆ F. Suppose that this inclusion morphism G → F is representable by open immersions, i. e., for any representable functor Hom (−, X ) and any morphism Hom (−, X )→ F, the fibered product G ×< sub > F </ sub > Hom (−, X ) is a representable functor Hom (−, Y ) and the morphism Y → X defined by the Yoneda lemma is an open immersion.

More precisely, find necessary and sufficient conditions on the tuple ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >) and ( y < sub > 1 </ sub >, ..., y < sub > n </ sub >) separately, so that there is an element of R with the property that x < sub > i </ sub >· r = y < sub > i </ sub > for all i. If D is the set of all R-module endomorphisms of U, then Schur's lemma asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the x's are linearly independent over D.

It can be verified that this is indeed a left module structure on U. As noted before, Schur's lemma proves D is a division ring if U is simple, and so U is a vector space over D.

The Schwarz – Pick lemma states that every holomorphic function on the unit disk U, or the upper half-plane H, with distances defined by the Poincaré metric, is a contraction mapping.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

Bézout's identity ( also called Bezout's lemma ) is a theorem in the elementary theory of numbers: let a and b be integers, not both zero, and let d be their greatest common divisor.

If a and b are coprime and a divides the product bc, then a divides c. This can be viewed as a generalization of Euclid's lemma.

Their intersection is, which can be shown to be non-context-free by the pumping lemma for context-free languages.

For regions in R < sup > 3 </ sup > more complicated than this, the latter statement might be false ( see Poincaré lemma ).

It can also be proven that none of these factors obeys Euclid's lemma ; e. g.

Euclid's classical lemma can be rephrased as " in the ring of integers every irreducible is prime ".

* choosing lemma forms for each word or part of word to be lemmatized

In the theory of formal languages in computability theory, a pumping lemma or pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of times, with the resulting string remaining in that language.

However, they cannot be used to determine if a language is in a given class, since satisfying the pumping lemma is a necessary, but not sufficient, condition for class membership.

Should there be a nontrivial factor dividing all the coefficients of the polynomial, then one can divide by the greatest common divisor of the coefficients so as to obtain a primitive polynomial in the sense of Gauss's lemma ; this does not alter the set of rational roots and only strengthens the divisibility conditions.

In the case of distinguishing between two models, each of which has no unknown parameters, use of the likelihood ratio test can be justified by the Neyman – Pearson lemma, which demonstrates that such a test has the highest power among all competitors.

The origin of her name is believed by some like Robert S. P. Beekes to be Pre-Greek and related to pēnelops ( πηνέλοψ ) or * pēnelōps (* πηνέλωψ ), glossed by Hesychius as " some kind of bird " ( today arbitrarily identified with the Eurasian Wigeon, to which Linnaeus gave the binomial Anas penelope ), where-elōps (- έλωψ ) is a common pre-Greek suffix for predatory animals ;< ref > Zeno. org lemma relating πηνέλωψ ( gen. πηνέλοπος ) and < χην ( ά ) λοπες >· ὄρνεα ( predators ) ποιά.

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function.

Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function.

The perianth is reduced to two scales, called lodicules, that expand and contract to spread the lemma and palea ; these are generally interpreted to be modified sepals.

and then note that the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed " S " shape of a slithering snake.

is a commutative diagram with exact rows, then the snake lemma can be applied twice, to the " front " and to the " back ", yielding two long exact sequences ; these are related by a commutative diagram of the form

To start with, none of the terms a, 2a, ..., ( p − 1 ) a can be congruent to zero modulo p, since if k is one of the numbers 1, 2, ..., p − 1, then k is relatively prime with p, and so is a, so Euclid's lemma tells us that ka shares no factor with p. Therefore, at least we know that the numbers a, 2a, ..., ( p − 1 ) a, when reduced modulo p, must be found among the numbers 1, 2, 3, ..., p − 1.

Although the Jacobi symbol can't be uniformly interpreted in terms of squares and non-squares, it can be uniformly interpreted as the sign of a permutation by Zolotarev's lemma.