## Are Noetherian rings finitely generated?

A ring is said to be Noetherian if every ideal in the ring is finitely generated. Right away we see that every principal ideal domain is a Noetherian ring since every ideal is generated by one element.

**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.

**Is polynomial ring finitely generated?**

It is finitely generated as an algebra, though.

### Can we prove every submodule of a Noetherian module is finitely generated How?

Proof. Considering R as a module over itself, we have that R is a Noetherian R- module and the ideal I is now a submodule of R. Since submodules and quotient of Noetherian modules are Noetherian (see Corollary 1.6), we have that R/I is a Noetherian R-module, i.e., every submodule of R/I is finitely generated.

**Is every ideal finitely generated?**

Therefore R is Noetherian. Definition An ideal I of a ring R is finitely generated if there is a finite subset A of R such that I = 〈A〉. Example Every principal ideal is finitely generated. Theorem A ring R is Noetherian if and only if every ideal of R is finitely generated.

**Why are Noetherian rings important?**

One big reason why they are important is that if R is noetherian the R[X] is also noetherian which then helps us see that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set (the solution set of a collection of polynomials in n …

#### 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 does finitely mean?**

having bounds or limits; not infinite; measurable. Mathematics. (of a set of elements) capable of being completely counted. not infinite or infinitesimal.

**Does every finitely generated module have a basis?**

In a field, every non-zero element has an inverse. This is no more true in a ring. However, in some rings, properties close to the existence of bases remain true. For instance, over a P.I.D., every submodule of a finitely generated free module is finitely generated and free.

## Is a submodule of a finitely generated module 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 Subrings of Noetherian rings Noetherian?**

If a commutative ring admits a faithful Noetherian module over it, then the ring is a Noetherian ring. (Eakin–Nagata) If a ring A is a subring of a commutative Noetherian ring B such that B is a finitely generated module over A, then A is a Noetherian ring.

**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.