A **finitely generated algebra** is a specific type of algebraic structure that is built from a vector space over a field (often denoted \( K \)) by introducing a multiplication operation. The key aspect of a finitely generated algebra is that it can be constructed using a finite number of generators. More formally, let \( A \) be a vector space over a field \( K \).
Articles by others on the same topic
A unital associative algebra over a field is finitely generated when a finite set 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 homomorphismso 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 .[ref]
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
Finite generation as an algebra differs from being finite-dimensional as a vector space. The polynomial ring has one algebra generator but infinitely many linearly independent monomials . By contrast, the quotient ring has basis , and its relation truncates every higher power.