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In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, an R-algebra A is finite if it is finitely generated as an R-
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
. An R-algebra can be thought as a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
of
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
f\colon R \to A, in this case f is called a finite morphism if A is a finite R-algebra. The definition of finite algebra is related to that of algebras of finite type.


Finite morphisms in algebraic geometry

This concept is closely related to that of
finite morphism In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is ...
in algebraic geometry; in the simplest case of
affine varieties In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ide ...
, given two affine varieties V\subset\mathbb^n, W\subset\mathbb^m and a dominant regular map \phi\colon V\to W, the induced homomorphism of \Bbbk-algebras \phi^*\colon\Gamma(W)\to\Gamma(V) defined by \phi^*f=f\circ\phi turns \Gamma(V) into a \Gamma(W)-algebra: : \phi is a finite morphism of affine varieties if \phi^*\colon\Gamma(W)\to\Gamma(V) is a finite morphism of \Bbbk-algebras. The generalisation to schemes can be found in the article on finite morphisms.


References


See also

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Finite morphism In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is ...
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Finitely generated algebra In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ''A'' over a field ''K'' where there exists a finite set of elements ''a''1,...,''a'n'' of ''A'' such that every element ...
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Finitely generated module In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts i ...
Commutative algebra Algebraic geometry Algebras {{Algebra-stub