Is submodule finitely generated?

In general, submodules of finitely generated modules need not be finitely generated. Since every polynomial contains only finitely many terms whose coefficients are non-zero, the R-module K is not finitely generated. In general, a module is said to be Noetherian if every submodule is finitely generated.

Is a Noetherian module finitely generated?

Every submodule of a noetherian module is noetherian. Since M is noetherian means every submodule N of M is finitely generated. Let L be a submodule of N then L is submodule of M. Thus, L is finitely generated and so N is noetherian module.

What is submodule in algebra?

A module over a ring that is contained in and has the same addition as another module over the same ring.

Are finitely generated modules free?

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.

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.

Is a submodule an ideal?

An Introduction to Homological Algebra 1. 0 and M are submodules of M; any submodule M′ ≠ M is called proper. 2. If M = R, its submodules are precisely the left ideals.

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.

What is a finitely generated subgroup?

A group such that all its subgroups are finitely generated is called Noetherian. A group such that every finitely generated subgroup is finite is called locally finite. Every locally finite group is periodic, i.e., every element has finite order. Conversely, every periodic abelian group is locally finite.

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.

Are submodules of free modules free?

every submodule of a free R-module is itself free; every ideal in R is a free R-module; R is a principal ideal domain.

Is 2Z a free module?

An important aspect of modules is that unlike vector spaces, modules are usually not free, i.e. they don’t have a basis. For example, take the Z-module given by Z/2Z. [ Recall: a Z-module is just the same as an abelian group, and a Z-module homomorphism corresponds precisely to a homomorphism of abelian groups. ]