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A commutative algebra over a field $k$ has a finite presentation if finitely many generators and finitely many polynomial relations describe it:The polynomial ring supplies freely chosen commuting generators, and the ideal of relations identifies expressions representing the same element. For example, $k[X,Y]/(XY-1)$ is the Laurent polynomial ring in one variable: $Y$ represents the multiplicative inverse of $X$.^[ref]

Every commutative finitely generated algebra over a field admits a finite presentation. Indeed, a field is a Noetherian ring; the Hilbert basis theorem makes its polynomial ring Noetherian, so the kernel of a chosen surjective algebra homomorphism is a finitely generated ideal. This is where commutativity enters the assertion. For a noncommutative associative algebra, a finite presentation instead uses a free associative algebra and finitely many generators for its two-sided ideal of relations; the commutative Hilbert basis theorem does not establish finite presentation there.^[ref]

A finite presentation also organizes differentiation: the module of Kähler differentials is generated by $d{X}_1,\dots ,d{X}_n$, with relations $d{f}_j={\sum }_i(\partial f_j/\partial X_i)d{X}_i$. The Jacobian matrix records those relations; this connects an algebraic description by equations to its infinitesimal geometry.