The Noetherian property for quotient rings of infinite polynomial rings

Abstract

Let <italic>X</italic> be an infinite set of indeterminates over the commutative Noetherian ring <italic>R</italic> with identity. We prove that the quotient ring of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R left-bracket upper X right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R[X]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to the multiplicative system of polynomials of unit content is also Noetherian. Moreover, we show that certain quotient rings of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R left-bracket upper X right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R[X]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to multiplicative systems of monic polynomials (where “monic” is appropriately defined) are also Noetherian.