Finitely-generated Algebra

Finitely-generated Algebra

In mathematics, a finitely generated algebra is an associative algebra A over a field K where there exists a finite set of elements a1,…,an of A such that every element of A can be expressed as a polynomial in a1,…,an, 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.

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Finitely-generated Algebra - Properties
... A homomorphic image of a finitely generated algebra is itself finitely generated ... theorem if A is a finitely generated commutative algebra then every ideal of A is finitely generated, or equivalently, A is a Noetherian ring ...

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