Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings

Abstract

. The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the S -Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the S -variant of Eakin–Nagata–Eisenbud theorem for general rings by using S -Noetherian modules. We also show that every right S -Noetherian domain is right Ore, which is embedded into a division ring. For a right S -Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right S -Noetherian or right Noetherian. As applications, the S -variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.