In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a finitely generated algebra (also called an algebra of finite type) is a
commutative associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
''A'' over a
field ''K'' where there exists a finite set of elements ''a''
1,...,''a''
''n'' of ''A'' such that every element of ''A'' can be expressed as a
polynomial in ''a''
1,...,''a''
''n'', with
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s in ''K''.
Equivalently, there exist elements
s.t. the evaluation homomorphism at
:
is
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
; thus, by applying the
first isomorphism theorem,
.
Conversely,
for any
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...