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.