Finitely generated algebra

Finitely generated algebra

In mathematics, a finitely generated algebra is an associative algebra "A" over a field "K" such that every element of "A" can be expressed as a polynomial in a finite set of elements "a"1,…,"a"n of "A", with coefficients in "K". If it is necessary to emphasize the field "K" then the algebra is said to be finitely generated over "K" . Algebras that are not finitely generated are called infinitely generated. Finitely generated commutative algebras are basic objects of consideration in modern algebraic geometry, where they correspond to affine algebraic varieties; for this reason, these algebras are also referred to as (commutative) affine algebras.

Examples

* The polynomial algebra "K" ["x"1,…,"x"n] is finitely generated. The polynomial algebra in countably many generators is infinitely generated.
* The field "E" = "K"("t") of rational functions in one variable over a given field "K" is "not" a finitely generated algebra over "K". On the other hand, "E" is generated over "K" by a single element, "t", "as a field".
* If "E" /"F" is a finite field extension then it follows from the definitions that "E" is a finitely generated algebra over "F".
* Conversely, if "E" /"F" is a field extension and "E" is a finitely generated algebra over "F" then the field extension is finite, see integral extension.
* If "G" is a finitely generated group then the group ring "KG" is a finitely generated algebra over "K".

Properties

* A homomorphic image of a finitely generated algebra is itself finitely generated. However, a similar property for subalgebras does not hold in general.
* Hilbert's basis theorem: if "A" is a finitely generated commutative algebra then every ideal of "A" is finitely generated, or equivalently, "A" is a Noetherian ring.

See also

* Finitely generated module


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Finitely generated — may refer to: * finitely generated group * finitely generated abelian group * finitely generated module: in particular, finitely generated ideal * finitely generated algebra * finitely generated space …   Wikipedia

  • Finitely-generated module — In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated R module also may be called a finite R module or finite over R.[1] Related concepts include finitely cogenerated modules, finitely… …   Wikipedia

  • Finitely-generated abelian group — In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. In this case, we …   Wikipedia

  • Finitely generated abelian group — In abstract algebra, an abelian group ( G ,+) is called finitely generated if there exist finitely many elements x 1,..., x s in G such that every x in G can be written in the form : x = n 1 x 1 + n 2 x 2 + ... + n s x s with integers n 1,..., n… …   Wikipedia

  • Structure theorem for finitely generated modules over a principal ideal domain — In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that… …   Wikipedia

  • Algebra (ring theory) — In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R .Any ring can be thought of as an algebra… …   Wikipedia

  • Affine algebra — may refer to: * affine Lie algebra, a type of Kac–Moody algebras * the Lie algebra of the affine group * finitely generated algebra …   Wikipedia

  • algebra, modern — ▪ mathematics Introduction also called  abstract algebra        branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers (real number), complex numbers (complex number), matrices (matrix), and… …   Universalium

  • Torsion (algebra) — In abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be free. Definition Let G be a group. An element g of G is called a torsion element if g has… …   Wikipedia

  • Exterior algebra — In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”