Are Noetherian rings finitely generated?

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.