= Finitely generated algebra
{wiki}
A unital <associative algebra> $A$ over a <field> $k$ is finitely generated when a <finite set> $a_1,\ldots,a_n$ generates it under <addition>, multiplication and <scalar multiplication>, including the <identity element>. Equivalently, every element is a finite <linear combination> of words in the generators. The words retain their order if multiplication is noncommutative.
In the commutative case, evaluating <polynomials> at the generators gives a <surjective> <algebra homomorphism>
$$
k[X_1,\ldots,X_n]\longrightarrow A,\qquad X_i\longmapsto a_i,
$$
so $A$ is a <quotient ring> of a <polynomial ring>. This equivalence assumes <commutativity>. For a noncommutative <associative algebra>, the corresponding source is the <free associative algebra> $k\langle X_1,\ldots,X_n\rangle$.<The Stacks project>[Commutative Algebra, §10.6, Definition 10.6.1, tag 00F2: finite type is defined by a polynomial-algebra surjection in the commutative setting]{ref}
Finite generation as an algebra differs from being finite-dimensional as a <vector space>. The <polynomial ring> $k[X]$ has one algebra generator but infinitely many <linearly independent> <monomials> $1,X,X^2,\ldots$. By contrast, the <quotient ring> $k[X]/(X^2)$ has <basis> $1,\bar X$, and its relation truncates every higher power.
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