The Noetherian property for quotient rings of infinite polynomial rings

Abstract:

Let X be an infinite set of indeterminates over the commutative Noetherian ring R with identity. We prove that the quotient ring of $R[X]$ with respect to the multiplicative system of polynomials of unit content is also Noetherian. Moreover, we show that certain quotient rings of $R[X]$ with respect to multiplicative systems of monic polynomials (where “monic” is appropriately defined) are also Noetherian.