Algebra (over a field)
Definition. Let $K$ be a field and let $R$ be a vector space over $K$ equipped with a bilinear product:
- $(x+y)\cdot z = x\cdot z+y\cdot z$
- $z\cdots (x+y)=z\cdot x+z\cdot y$
- $(ax)\cdot (by)=(ab)(x\cdot y)$ Then $R$ is called an algebra.
Definition. An algebra $R$ over $K$ is said to be finitely generated if there exist $f_1,\dots,f_n\in R$ such that any $a\in R$ is the evaluation of a degree $n$ polynomial in $K$ at the basis elements, i.e. $a=F(f_1,\dots,f_n)$ where $F\in K[x_1,\dots,x_n]$.
Corollary. If $R$ is finitely generated as an algebra, then it is also a Noetherian ring.