Endo-Noetherian Skew Generalized Power Series Rings

Document Type : Novel Research Articles

Authors

1 Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt.

2 Mathematics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt

Abstract

Endo-Noetherian modules were introduced by A. Kaidi and E. Sanchez] as a generalization of Noetherian modules. A left Ɍ-module M which satisfies the ascending chain condition for endomorphic kernels is said to be endo-Noetherian. A ring Ɍ is left endo-Noetherian if Ɍ over itself as a left module is endo-Noetherian. The property of endo-Noetherian were studied for the polynomial ring Ɍ[x] and the formal power series ring Ɍ[[x]]. Throughout this article, we are interested in the study of the left endo-Noetherian property of the skew generalized power series ring Ɍ[[ℵ, σ]], we show under what conditions on a ring Ɍ, a strictly ordered monoid (ℵ, ≼), and a monoid homomorphism σ: ℵ ⟶ End (Ɍ), the skew generalized power series ring Ɍ[[ℵ, σ]] is left endo-Noetherian if and only if Ɍ is left endo-Noetherian. We find new corollaries on generalized power series rings, power series rings, and polynomial rings as special examples of our general conclusion.

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