Hilbert's basis theorem

Hilbert's Basis Theorem is a fundamental result in algebra, particularly in the theory of rings and ideals. It states that if \( R \) is a Noetherian ring (meaning that every ideal in \( R \) is finitely generated), then any ideal in the polynomial ring \( R[x] \) (the ring of polynomials in one variable \( x \) with coefficients in \( R \)) is also finitely generated.