What is meant by finitely generated?
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 finite algebra finitely generated?
No, being finitely generated as an algebra is generally not as strong as being finitely generated as a module. Being finitely generated as an algebra means that there is some finite set of elements from the algebra, such that the subalgebra generated by those elements is the entire algebra.
What is finitely generated ideal?
An ideal in a ring is called finitely generated if and only if it can be generated by a finite set. An ideal is called principal if and only if it can be generated by a single element. Definition. A ring R is Noetherian if and only if all of its ideals are finitely gener- ated.
Is a polynomial ring finitely generated?
For example, a polynomial ring R[x] is finitely generated by {1, x} as a ring, but not as a module. If A is a commutative algebra (with unity) over R, then the following two statements are equivalent: A is a finitely generated R module. A is both a finitely generated ring over R and an integral extension of R.
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. .
Are the integers finitely generated?
Consider the set of integers Z together with the regular addition operation. This group itself is infinite (you can always keep adding 1), but since a finite set such as {1} can generate it, it’s considered finitely generated.
What is a finitely generated field extension?
An extension K/k is said to be finitely generated (or an extension of finite type) if there is a finite subset S of k such that K coincides with the smallest subfield containing S and k. In this case one says that K is generated by S over k.
Are all algebraic extensions finite?
5. A finite extension is algebraic. In fact, an extension E/k is algebraic if and only if every subextension k(\alpha )/k generated by some \alpha \in E is finite. In general, it is very false that an algebraic extension is finite.
Is every Noetherian ring finitely generated?
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.
What is the generator of an ideal?
An ideal of the form (a) is called a principal ideal with generator a. We have b ∈ (a) if and only if a | b. Note (1) = R. An ideal containing an invertible element u also contains u−1u = 1 and thus contains every r ∈ R since r = r · 1, so the ideal is R.
Is Q finitely generated?
The sets C, R, and Q are NOT finitely generated. The set of complex and real numbers are both uncountable, and any finitely generated group must be countable (it’s a Foundations of Mathematics style proof).
Is a finitely generated abelian group finite?
Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.
What is field extension in algebra?
In mathematics, particularly in algebra, a field extension is a pair of fields. such that the operations of E are those of F restricted to E. In this case, F is an extension field of E and E is a subfield of F.
What is relation between extension field and vector space?
Any extension field K of F is a vector space over F, using the addition operation in K and multiplication in K as the scalar multiplication. (All the properties are immediate, except that 1v=v for all v\in K. To be sure of that, we must know that the identity of F is the identity of K.
What is meant by finite extension?
A finite extension is an extension that has a finite degree. Given two extensions L / K and M / L, the extension M / K is finite if and only if both L / K and M / L are finite. In this case, one has. Given a field extension L / K and a subset S of L, there is a smallest subfield of L that contains K and S.
What is algebraic and transcendental extension?
ALGEBRAIC AND TRANSCENDENTAL EXTENSIONS. Definition: A field extension E/F is said to be an algebraic extension, and E is said to be algebraic over F, if all elements of E are algebraic over F. Otherwise, E is transcendental over F. Thus, E/F is transcendental if at least one element of E is transcendental over F.
What does Noetherian mean?
In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite length.
What is Noetherian induction?
From Encyclopedia of Mathematics. A reasoning principle applicable to a partially ordered set in which every non-empty subset contains a minimal element; for example, the set of closed subsets in some Noetherian space.
Is every ideal finitely generated?
Example Every principal ideal is finitely generated. Theorem A ring R is Noetherian if and only if every ideal of R is finitely generated.
What is an ideal in algebra?
ideal, in modern algebra, a subring of a mathematical ring with certain absorption properties. The concept of an ideal was first defined and developed by German mathematician Richard Dedekind in 1871. In particular, he used ideals to translate ordinary properties of arithmetic into properties of sets.