Hilbert's basis theorem

In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.

If ${\displaystyle R}$ a ring, let ${\displaystyle R[X]}$ denote the ring of polynomials in the indeterminate ${\displaystyle X}$ over ${\displaystyle R}$. Hilbert proved that if ${\displaystyle R}$ is "not too large", in the sense that if ${\displaystyle R}$ is Noetherian, the same must be true for ${\displaystyle R[X]}$. Formally,

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