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Rings Whose Non-Invertible Elements Are Uniquely Strongly Clean

Peter Danchev, Omid Hasanzadeh, Ahmad Moussavi

Abstract

We define and explore in details the class of GUSC rings, that are those rings whose non-invertible elements are uniquely strongly clean. These rings are a common generalization of the so-called USC rings, introduced by Chen-Wang-Zhou in J. Pure & Appl. Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also generalize the so-called GUC rings, defined by Guo-Jiang in Bull. Transilvania Univ. Braşov (2023), which are rings whose non-invertible elements are uniquely clean.

Rings Whose Non-Invertible Elements Are Uniquely Strongly Clean

Abstract

We define and explore in details the class of GUSC rings, that are those rings whose non-invertible elements are uniquely strongly clean. These rings are a common generalization of the so-called USC rings, introduced by Chen-Wang-Zhou in J. Pure & Appl. Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also generalize the so-called GUC rings, defined by Guo-Jiang in Bull. Transilvania Univ. Braşov (2023), which are rings whose non-invertible elements are uniquely clean.
Paper Structure (3 sections, 26 theorems, 21 equations)

This paper contains 3 sections, 26 theorems, 21 equations.

Table of Contents

  1. Introduction and Basic Concepts
  2. Examples and Main Results
  3. Group Rings

Key Result

Lemma 2.2

An element $a$ in a ring $R$ is USC if, and only if, so is $1-a$.

Theorems & Definitions (59)

  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 49 more