## When a module is called finitely generated?

An A-module M is said to be finitely generated if it has a finite set of generators. Examples: 1) Any finite dimensional vector space over a field k is a finitely generated k-module. 2) Any finitely generated abelian group is a finitely generated Z-module.

## What does it mean for a ring to be finitely generated?

A ring is an associative algebra over the integers, hence a ℤ-ring. Accordingly a finitely generated ring is a finitely generated ℤ-algebra, and similarly for finitely presented ring. For rings every finitely generated ring is already also finitely presented.

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.

### Are Noetherian rings finitely generated?

Since the domain is Noetherian, the codomain is Noetherian as well. This is a Noetherian ring, but it is not finitely generated, because there are infinitely many primes.

### How do you prove a module is finitely generated?

A module M is finitely generated if and only if any increasing chain Mi of submodules with union M stabilizes: i.e., there is some i such that Mi = M. This fact with Zorn’s lemma implies that every nonzero finitely generated module admits maximal submodules.

What is the rank of a module?

The rank of a free module M over an arbitrary ring R( cf. Free module) is defined as the number of its free generators. For rings that can be imbedded into skew-fields this definition coincides with that in 1). In general, the rank of a free module is not uniquely defined.

## 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.

## Are submodules of free modules free?

Submodules of free modules 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 every Noetherian ring Artinian?

A ring A is noetherian, respectively artinian, if it is noetherian, respectively artinian, considered as an A-module. In other words, the ring A is noetherian, respectively artinian, if every chain a1 ⊆ a2 ⊆ ··· of ideal ai in A is stable, respectively if every chain a1 ⊇ a2 ⊇··· of ideals ai in A is stable.

### Is a PID a Noetherian ring?

Thus every PID is a Noetherian integral domain. Proof. In a principal ideal ring R, every left or right ideal is generated by a single element and hence in particular, it is finitely generated. Thus R is a Noetherian ring by the Theorem 1.4.

### Is the Z-module Q finitely generated?

Q is obviously a Z-module, however, it is not finitely generated.

Which of the following module is not free module?

The submodule 2Z/4Z is not free. For an explicit counterexample, consider R=k[X,Y] for any field k, which is free over itself. Then the ideal m=RX+RY is not free.

## 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).

## What kind of module has a finite generating set?

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 .

When is a finitely generated module over a PID free?

A finitely generated module over a principal ideal domain is torsion-free if and only if it is free. This is a consequence of the structure theorem for finitely generated modules over a principal ideal domain, the basic form of which says a finitely generated module over a PID is a direct sum of a torsion module and a free module.

### 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.