Finite algebra

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (January 2020) (Learn how and when to remove this template message)

|

In abstract algebra, an associative algebra A over a ring R is called finite if it is finitely generated as an R-module. An R-algebra can be thought as a homomorphism of rings f:R->A, in this case f is called a finite morphism if A is a finite R-algebra.[1]

Being a finite algebra is a stronger condition than being an algebra of finite type.

## Finite morphisms in algebraic geometry

This concept is closely related to that of finite morphism in algebraic geometry; in the simplest case of affine varieties, given two affine varieties V⊆𝔸n, W⊆𝔸m and a dominant regular map ϕ:V->W, the induced homomorphism of k-algebras ϕ*:Γ(W)->Γ(V) defined by ϕ*f=f∘ϕ turns Γ(V) into a Γ(W)-algebra:

ϕ is a finite morphism of affine varieties if ϕ*:Γ(W)->Γ(V) is a finite morphism of k-algebras.[2]

The generalisation to schemes can be found in the article on finite morphisms.

## References

1. ↑ Atiyah, Michael Francis; Macdonald, Ian Grant (1994). Introduction to commutative algebra. CRC Press. p. 30. ISBN 9780201407518. https://www.crcpress.com/Introduction-To-Commutative-Algebra/Atiyah/p/book/9780201407518.
2. ↑ Perrin, Daniel (2008). Algebraic Geometry An Introduction. Springer. p. 82. ISBN 978-1-84800-056-8. https://www.springer.com/gp/book/9781848000551.

## See also

* Finite morphism
* Finitely generated algebra
* Finitely generated module

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Finite algebra. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Finite_algebra&oldid=3986668"