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Longer titles found: Structure theorem for finitely generated modules over a principal ideal domain (view)

searching for Finitely generated module 14 found (77 total)

alternate case: finitely generated module

Analytically unramified ring (557 words) [view diff] exact match in snippet view article find links to article

field of fractions K of R, the integral closure of S in K is a finitely generated module over S. The second follows from the first. Let K0 be a perfect

Iwasawa algebra (1,182 words) [view diff] exact match in snippet view article find links to article

divisible by p). Height 2: the maximal ideal (p,T). The rank of a finitely generated module is the number of times the module Zp[[T]] occurs in it. This is

Free module (1,808 words) [view diff] exact match in snippet view article find links to article

modules is generally not free (cf. the Baer–Specker group). A finitely generated module over a commutative local ring is free if and only if it is faithfully

Torsion-free module (597 words) [view diff] no match in snippet view article find links to article

isomorphic to an ideal of the ring. Over a Dedekind domain, a finitely-generated module is torsion-free if and only if it is projective, but is in general

Koszul complex (5,545 words) [view diff] exact match in snippet view article find links to article

holds. More generally, Theorem— Let R be a ring and M a nonzero finitely generated module over R . If x 1 , x 2 , … , x r {\displaystyle x_{1},x_{2},\dots

Essential extension (975 words) [view diff] no match in snippet view article find links to article

lemma is that J(R)M is a superfluous submodule of M when M is a finitely-generated module over R. This definition can be generalized to an arbitrary abelian

Completion of a ring (1,582 words) [view diff] exact match in snippet view article find links to article

Noetherian ring R is a flat module over R. The completion of a finitely generated module M over a Noetherian ring R can be obtained by extension of scalars:

Krull's principal ideal theorem (1,235 words) [view diff] exact match in snippet view article find links to article

{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 M=IM} for some ideal I contained

Affine monoid (1,113 words) [view diff] exact match in snippet view article find links to article

{\displaystyle \mathbb {Q} ^{n}} . Then P ∩ L {\displaystyle P\cap L} is a finitely generated module over the affine monoid C ∩ L {\displaystyle C\cap L} . (Theorem

Bilinear form (2,727 words) [view diff] exact match in snippet view article find links to article

bilinear form is unimodular if V → V∗ is an isomorphism. Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate"

Proj construction (3,582 words) [view diff] no match in snippet view article find links to article

x {\displaystyle O_{X,x}} then the degree-one elements form a finitely-generated module over O X , x {\displaystyle O_{X,x}} and also generate the stalk

Ext functor (3,876 words) [view diff] exact match in snippet view article find links to article

}\operatorname {Ext} _{R}^{i}(M,N_{\alpha })\end{aligned}}} Let A be a finitely generated module over a commutative Noetherian ring R. Then Ext commutes with localization

Weyl algebra (4,164 words) [view diff] exact match in snippet view article find links to article

so the Weyl algebra has a very large center. In fact, it is a finitely generated module over its center; even more so, it is an Azumaya algebra over its

List of inventions and discoveries by women (7,598 words) [view diff] exact match in snippet view article find links to article

algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over the polynomial ring S:=k[y1, y2, ..., yd]. The theorem has