Is a finitely generated ring Noetherian?
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. Well, in short, “Noetherian-ness” is a property which generalizes “PID-ness”.
Are local rings Noetherian?
Complete Noetherian local rings are classified by the Cohen structure theorem. In algebraic geometry, especially when R is the local ring of a scheme at some point P, R / m is called the residue field of the local ring or residue field of the point P.
What does it mean for a ring to be finitely generated?
An ideal in a commutative unital ring is said to be finitely generated if it has a finite generating set, that is, if there is a finite set such that it is the smallest ideal containing that finite set.
Is every ring 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 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.
Are polynomial rings Local?
The ring of formal power series k[[X1…Xn]] over a field k or over any local ring is local. On the other hand, the polynomial ring k[X1…Xn] with n≥1 is not local. The ring Ap, which consists of fractions of the form a/s, where a∈A, s∈A∖p, is local and is called the localization of the ring A at p.
Is rational numbers finitely generated?
The Group of Rational Numbers is Not Finitely Generated.
What is a finitely generated field extension?
Let E/F be a field extension. Then E is said to be finitely generated over F if and only if, for some α1,…,αn∈E: E=F(α1,…,αn) where F(α1,…,αn) is the field in E generated by F∪{α1,…,αn}.