A Note on Noetherian Polynomial Modules

초록

Let R be a commutative ring and let M be an R-module. In this note, we give a brief proof of the Hilbert basis theorem for Noetherian modules. This states that if R contains the identity and M is a Noetherian unitary R-module, then M[X] is a Noetherian R[X]-module. We also show that if M[X] is a Noetherian R[X]-module, then M is a Noetherian R-module and there exists an element e is an element of R such that em = m for all m is an element of M. Finally, we prove that if M[X] is a Noetherian R[X]-module and ann(R)(M) = (0), then R has the identity and M is a unitary R-module.

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