HilbertBasisSatz

HilbertBasisSatz is a fundamental theorem in abstract algebra, specifically in the area of commutative algebra. It states that if R is a commutative ring with unity, and if M is an R-module that is finitely generated, then every submodule of M is also finitely generated. A direct consequence of this theorem, and perhaps its most common application, is that if R is a Noetherian ring, then the polynomial ring R[x] is also a Noetherian ring. A ring is Noetherian if every ideal in the ring is finitely generated. This means that any ideal in R[x] can be expressed as the span of a finite set of polynomials. The Hilbert Basis Theorem is crucial for understanding the structure of polynomial rings and their ideals. It provides a powerful tool for proving many other results in algebraic geometry and commutative algebra. For example, it is used to show that the ring of invariants of a finite group acting on a polynomial ring is also a Noetherian ring. The proof of the Hilbert Basis Theorem itself typically involves an inductive argument.