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Notes on Reversed Products of Two Elements in Rings
Abstract
This paper investigates reversed product properties of two elements in rings. Motivated by Cline's formula, we characterize reversible and ∗-reversible rings in terms of group invertible elements, EP elements, and the transfer behaviour of generalized inverses for reversed products. We prove that a unital ring R is reversible if and only if ab∈ R♯ yields ba∈ R♯. For an involutive ring R, R is ∗-reversible precisely whenever ab∈ REP implies b∗a∈ REP. Several counterexamples are constructed to differentiate these ring classes, and the mutual inclusion relations among them are also discussed.
Related
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BibTeX
@misc{wang2026notes,
title = {{Notes on Reversed Products of Two Elements in Rings}},
author = {Long Wang and Tingting Li and Yuheng Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12643},
eprint = {2609.12643},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12643},
}
RIS
TY - UNPB
TI - Notes on Reversed Products of Two Elements in Rings
AU - Long Wang
AU - Tingting Li
AU - Yuheng Liu
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12643
UR - https://arxiv.org/abs/2609.12643
AB - This paper investigates reversed product properties of two elements in rings. Motivated by Cline's formula, we characterize reversible and ∗-reversible rings in terms of group invertible elements, EP elements, and the transfer behaviour of generalized inverses for reversed products. We prove that a unital ring R is reversible if and only if ab∈ R^♯ yields ba∈ R^♯. For an involutive ring R, R is ∗-reversible precisely whenever ab∈ R^EP implies b^∗a∈ R^EP. Several counterexamples are constructed to differentiate these ring classes, and the mutual inclusion relations among them are also discussed.
ER -