An algebra that is finite dimensional as a vector space over a field is a Noetherian ring

Given: A field $k$, which is in particular a commutative ring (Every field is a commutative ring with $1\mathrm{\ne }0$; it is an integral domain, and it is a commutative division ring), and a commutative $k$-algebra $A$ with structure map ${\eta }_A\text{\hspace{0.17em}⁣}:k\to A$ whose underlying $k$-vector space is finite dimensional of dimension $n$.

An $R$-algebra is a unital ring $A$ with a unital ring homomorphism ${\eta }_A\text{\hspace{0.17em}⁣}:R\to A$ of central image; the induced scalar action $ra:={\eta }_A(r)a$ makes $A$ an $R$-module (Algebras over a commutative ring, central structure maps, and algebra homomorphisms).

A vector space over a field $F$ is a set $V$ with an addition making $(V,+,0_V)$ an abelian group and a scalar multiplication $F\times V\to V$ satisfying $\lambda (u+v)=\lambda u+\lambda v$, $(\lambda +\mu )v=\lambda v+\mu v$, $(\lambda \mu )v=\lambda (\mu v)$ and $1_{F}v=v$ (Vector space over a field).

A linear subspace of a vector space $V$ over $F$ is a subset containing $0_V$ and closed under addition and under scalar multiplication, and it is itself a vector space over $F$ under the restricted operations (Linear subspace of a vector space).

$V$ is finite-dimensional over $F$ when it has a finite basis, and ${\dim }_{F}V$ is the unique $n\in \mathbb{N}$ with a basis $B$ satisfying $B\approx n$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis).

If $V$ is finite dimensional over $F$ with ${\dim }_{F}V=n$ and $U$ is a linear subspace of $V$, then $U$ is finite dimensional over $F$ and ${\dim }_{F}U\le n$ (If ${\dim }_{F}V=n$ and $U$ is a linear subspace of $V$, then $U$ is finite-dimensional, ${\dim }_{F}U\le n$, and ${\dim }_{F}U=n$ if and only if $U=V$).

A subset $B\subseteq V$ is a basis when it is linearly independent and spans $V$ (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis).

The span of a subset is the set of its finite linear combinations, and a subset spans $V$ when its span is $V$ (Linear combination of a finite list, and the span $\mathrm{span}(S)$ as the smallest linear subspace containing $S$).

In a commutative ring, $(S)$ consists of finite sums $\sum r_i{s}_i$, and $(a)=Ra$ (In a commutative ring, $(S)$ consists of finite sums $\sum r_i{s}_i$, and $(a)=Ra$).

For a commutative ring, being Noetherian is equivalent to every ideal being finitely generated (A commutative ring is Noetherian exactly when every ideal is finitely generated, exactly when its ideals satisfy the ascending chain condition, and exactly when every nonempty set of ideals has a maximal member).

An algebra is module-finite over $R$ when it is finitely generated as an $R$-module (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras).

A module-finite commutative algebra over a Noetherian commutative ring is a Noetherian ring (A module-finite algebra over a Noetherian ring is a Noetherian ring, and so is every ring between the two).

Every field is a Noetherian ring (Fields and $\mathbb{Z}$ are Noetherian, and so are their polynomial rings in finitely many variables).