On Some Localizations and Their Relation to Quotients

Abstract

We describe the ring of fractions of $\mathbb{Z}/n\mathbb{Z}$ with respect to the multiplicative set $S=\{1,x,x^2,x^3,...\}$, where x is any non-nilpotent element and we observe how it identifies with a quotient of the ring. Given a ring $R=\mathbb{Z}/n\mathbb{Z}$ and a multiplicative set S, we characterize all ideals I of R such that R/I is isomorphic to $S^{-1}R$ under the natural map.

MSC:

• 13B30

Acknowledgment

We would like to thank both the referees for their comments and showing us how to complete the proof of Lemma 3, and the editor for her helpful suggestions. We are grateful to Katrina Honigs for her valuable inputs.

Additional information

Notes on contributors

Nawal Kishor Hazarika

Nawal Kishor Hazarika is a final year student of the Integrated M.Sc. program in Mathematics at Tezpur University, Assam. He is interested in Commutative Algebra and Abstract Algebra. Department of Mathematical Sciences, Tezpur University, Napaam, Assam 784028, INDIA

e-mail: [email protected]

Pijush Pratim Sarmah

Pijush Pratim Sarmah is pursuing a M.Sc in Mathematics at Simon Fraser University, Burnaby. His interests are in Algebraic and Arithmetic Geometry. Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada

e-mail: [email protected]