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 [\!

Are locally free modules projective?

A projective module over a local ring is free. Thus a projective module is locally free (in the sense that its localization at every prime ideal is free over the corresponding localization of the ring).

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.

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.

Is Z torsion free?

Both these examples can be generalized as follows: if R is a commutative domain and Q is its field of fractions, then Q/R is a torsion R-module. The torsion subgroup of (R/Z, +) is (Q/Z, +) while the groups (R, +) and (Z, +) are torsion-free.

How do you prove a module is flat?

Faithful flatness A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Although the concept is defined for modules over a non-necessary commutative ring, it is used mainly for commutative algebras.

What is torsion free ring?

In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. A module is torsion free if its torsion submodule comprises only the zero element.

Are free modules torsion free?

In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is torsion free if its torsion submodule is reduced to its zero element.

How can I see what modules are free?

In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.

Is the projective module over a local ring free?

In this section we prove a very cute result: a projective module over a local ring is free (Theorem 10.85.4 below). Note that with the additional assumption that is finite, this result is Lemma 10.78.5. In general we have:

Is the localization of a projective module locally free?

Thus a projective module is locally free (in the sense that its localization at every prime ideal is free over the corresponding localization of the ring). The converse is true for finitely generated modules over Noetherian rings: a finitely generated module over a commutative noetherian ring is locally free if and only if it is projective.

Is the category of finitely generated projective modules an exact category?

The category of finitely generated projective modules over a ring is an exact category. (See also algebraic K-theory ). Given a module, M, a projective resolution of M is an infinite exact sequence of modules with all the Pi s projective. Every module possesses a projective resolution. In fact a free resolution (resolution by free modules) exists.

Which is the simplest example of a projective module?

Projective modules with finitely many generators are studied in algebraic $ K $- theory. The simplest example of a projective module is a free module. Over rings decomposable into a direct sum there always exist projective modules different from free ones.