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Are projective modules finitely generated?

Are projective modules finitely generated?

P is a finitely generated projective module, i.e. there exist an integer n, an \mathbf {A}-module N and an isomorphism of P\oplus N over \mathbf {A}^{\! n} . There exist an integer n, elements (g_i)_{i\in [\! P is finitely generated, and for every finite system of generators (h_i)_{i\in [\!

What is a finitely presented module?

From Wikipedia, the free encyclopedia. In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type.

Is R 0 finitely generated?

If we consider R А R asa R module it too is finitely generated with a generating set { ( 1 , 0), (0, 1 ) } .

Is free module finitely generated?

A finitely generated torsion-free module of a commutative PID is free. A finitely generated Z-module is free if and only if it is flat. See local ring, perfect ring and Dedekind ring.

Is 0 a projective module?

In general though, projective modules need not be free: Over a direct product of rings R × S where R and S are nonzero rings, both R × 0 and 0 × S are non-free projective modules. More generally, over any semisimple ring, every module is projective, but the zero ideal and the ring itself are the only free ideals.

What is a locally free module?

Definition An R-module N over a Noetherian ring R is called a locally free module if there is a cover by ideals I↪R such that the localization NI is a free module over the localization RI.

What is meant by finitely generated?

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.

Does every finitely generated module have a basis?

I know that a free module is a module with a basis, and that a finitely generated module has a finite set of generating elements (ie any element of the ring can be expressed as a linear combination of those generators). …

What is meant by finitely generated Abelian groups?

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set of or that generate . Every finite abelian group is finitely generated.

Is Z projective Z module?

Z2 is not projective as a Z module. Since every projective module is flat (projective modules are direct summands of free modules, free modules are flat, and direct summands of flat modules are flat), it suffices to show that Z2 is not flat as a Z module.

Is Z 6Z a free module?

2) Z/2Z and Z/3Z are non-free projective Z/6Z-modules. be a sequence of R-modules and R-module homomorphisms.

What is a finitely generated vector space?

A vector space V is said to be finitely generated if there exists a finite set S such that Span (S) = V . It is easy to see that a finitely generated vector space has a basis (which is a subset of the generating set S).

When is a module a finitely presented module?

Informally, is a finitely presented -module if and only if it is finitely generated and the module of relations among these generators is finitely generated as well. A choice of an exact sequence as in the definition is called a presentation of . Lemma 10.5.2. Let be a ring.

What is the generic rank of a finitely generated module?

A finitely generated projective module over a Noetherian integral domain has constant rank and so the generic rank of a finitely generated module over A is the rank of its projective part. The following conditions are equivalent to M being finitely generated (f.g.):

How is the left R-module M finitely generated?

The left R -module M is finitely generated if there exist a1, a2., an in M such that for any x in M, there exist r1, r2., rn in R with x = r1a1 + r2a2 + + rnan . The set { a1, a2., an } is referred to as a generating set of M in this case. A finite generating set need not be a basis, since it need not be linearly independent over R.

How are finitely generated modules over a division ring classified?

Finitely generated modules over the ring of integers Z coincide with the finitely generated abelian groups. These are completely classified by the structure theorem, taking Z as the principal ideal domain. Finitely generated (say left) modules over a division ring are precisely finite dimensional vector spaces (over the division ring).

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Ruth Doyle