International Journal of Nonlinear Analysis and Applications

Weakly unit regular clean rings

Articles in Press

Document Type : Research Paper

Author

Department of Mathematics, Technical and Vocational University (TVU), Tehran, Iran

Abstract

A ring $R$ is called clean if every member of $R$ is the sum of a self-resolved member and an invertible member. Also, we call the ring $R$ weakly clean if every member of $R$ can be written as the sum or difference of an invertible member and an autoregressive member. The $a\in R$ member is called unit regular whenever $u\in U(R)$ exists such $a=aua$. A ring $R$ is called a clean unit if every member of $R$ is the sum of an autonomial member and a unitary member. We call a ring $R$ a weakly clean unit if every member of $R$ can be written as the sum or difference of a unit regular and a self-power term. In this paper, weakly unit regular clean rings are introduced and discussed. In particular, we show that if $\{R_i\}_{i\in I}$ is a family of commutative rings, then $R=\prod_{i\in I} R_i$ is weakly unit regular clean if and only if every $R_i$ is weakly clean regular unit and at most one $R_i$'s are not clean and regular units.

Keywords

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International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 10 June 2025
  • Receive Date: 16 March 2024
  • Accept Date: 04 May 2024