mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction...

conclusion. This is essentially the statement of Nakayama's lemma. Precisely, one has the following. Nakayama's lemma: Let U be a finitely generated right module...

local lemma Nakayama's lemma Poincaré's lemma Riesz's lemma Schur's lemma Schwarz's lemma Sperner's lemma Urysohn's lemma Vitali covering lemma Yoneda's...

{\displaystyle N=IN} . Thus, if A is local, N = 0 {\displaystyle N=0} by Nakayama's lemma. If A is an integral domain, then one uses the determinant trick (that...

prominent role in many ring- and module-theoretic results, such as Nakayama's lemma. There are multiple equivalent definitions and characterizations of...

Hausdorff space. The theorem is a consequence of the Artin–Rees lemma together with Nakayama's lemma, and, as such, the "Noetherian" assumption is crucial. Indeed...

ideal m and residue field k and M finitely generated module. Then Nakayama's lemma says that M has a minimal generating set whose cardinality is dim k...

particular of Lasker–Noether theorem, the Krull intersection theorem, and Nakayama's lemma. Furthermore, if a ring is Noetherian, then it satisfies the descending...

over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma. For the general case, the proof (both the original as well as later...

{q}}^{(n+1)}=(x){\mathfrak {q}}^{(n)}/{\mathfrak {q}}^{(n+1)}} . Then, by Nakayama's lemma (which says a finitely generated module M is zero if M = I M {\displaystyle...

University, and Hamburg University. Nakayama's lemma, Nakayama algebras, Nakayama's conjecture and Murnaghan–Nakayama rule are named after him. Scholia...

Abhyankar's lemma Fundamental lemma (Langlands program) Five lemma Horseshoe lemma Nine lemma Short five lemma Snake lemma Splitting lemma Yoneda lemma Matrix...

Nakayamadaira-Onsen Station Nakayama lemma, a lemma in commutative algebra This disambiguation page lists articles associated with the title Nakayama. If an internal...

associated to M.) Moreover, if R is a Noetherian integral domain, then, by Nakayama's lemma, these conditions are equivalent to The dimension of the k ( p ) {\displaystyle...

finitely generated modules over a commutative ring R, Nakayama's lemma is fundamental. Sometimes, the lemma allows one to prove finite dimensional vector spaces...

and right primitive ideals. The following simple but important fact (Nakayama's lemma) is built-in to the definition of a Jacobson radical: if M is a module...

operation is matrix multiplication. General Isomorphism theorems for rings Nakayama's lemma Structure theorems The Artin–Wedderburn theorem determines the structure...

x ′ ; M ) ) {\displaystyle N=\operatorname {H} _{1}(K(x';M))} . By Nakayama's lemma, N = 0 {\displaystyle N=0} and so x' is a regular sequence by the inductive...

with pd R ⁡ M = 1 {\displaystyle \operatorname {pd} _{R}M=1} . By Nakayama's lemma, there is a surjection F → M {\displaystyle F\to M} from a free module...

Fitting lemma Schur's lemma Nakayama's lemma Krull–Schmidt theorem Steinitz exchange lemma Jordan–Hölder theorem Artin–Rees lemma Schanuel's lemma Morita...

multiplication by b) gives (iv) ⇒ (i) and the rest is easy. Coincidentally, Nakayama's lemma is also an immediate consequence of this theorem. It follows from the...

cohomology or relative homological algebra, Shapiro's lemma, also known as the Eckmann–Shapiro lemma, relates extensions of modules over one ring to extensions...

its characteristic polynomial relative to the generators of M; see Nakayama's lemma#Proof. If ϕ {\displaystyle \phi } is surjective, then it is injective...

x} , more than would be true for an arbitrary sheaf. For example, Nakayama's lemma says (in geometric language) that if F {\displaystyle {\mathcal {F}}}...

Japanese ring. These are also called pseudo-geometric rings. Nakayama's lemma Nakayama's lemma states that if a finitely generated module M is equal to IM...

k = R / m. Any R-module M yields a k-vector space given by M / mM. Nakayama's lemma shows this passage is preserving important information: a finitely...

lies in JFi − 1, where J is the irrelevant ideal. As a consequence of Nakayama's lemma, φ then takes a given basis in Fi to a minimal set of generators in...

Prop. 2.4) (which in fact is the more general statement related to the Nakayama lemma; one takes for the ideal in that proposition the whole ring R). The...

and only if m ≤ r {\displaystyle m\leq r} . The proof follows from Nakayama's lemma. There has been extensive study into the other containment, when symbolic...

easier to study than general commutative rings, in particular because of Nakayama lemma. However, the main reason is that many properties are true for a ring...