A -algebra is finitely generated when finitely many elements generate it under addition, multiplication, and scalar multiplication. Equivalently, it is a quotient of a polynomial ring .
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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 \( K \).