Finitely Generated Algebra

A finitely generated algebra over a commutative ring with multiplicative identity, in the commutative convention used here, is a unital commutative algebra over with a finite list of elements , ..., such that every element of is a polynomial expression in them with coefficients from the image of the structure ring homomorphism (Milne 2014, p. 3). Equivalently, it is isomorphic as an -algebra to a quotient ring of a polynomial ring in finitely many indeterminates, so

for some ideal . Such an algebra is also called an algebra of finite type over .

Every finite algebra is finitely generated, but the converse fails. For a field , the polynomial ring is generated as an algebra by , but its vector space dimension over is infinite, so it is not a finite algebra. In the extension ring case, , the notation means adjoining these elements as ring generators, not merely taking their -linear combinations.