Transposed Triple Products and Pro-Symmetric Rings in ∗-Rings
Abstract
This paper investigates properties concerning transposed triple products in rings. Motivated by Cline's formula, we characterize symmetric rings by means of group invertible elements and EP elements. We prove that a unital ring R is symmetric if and only if abc∈ R♯ implies acb∈ R♯ for all a,b,c∈ R. In particular, we give an answer to the problem posed in \cite[Problem 2.9]{MW1}. An example is provided to illustrate that for a symmetric ring R, abc=e does not generally yield acb=e. For ∗-rings, we introduce the notion of pro-symmetric rings: a ring R is pro-symmetric if abc∈ P(R) implies acb∈ P(R) for all a,b,c∈ R. We show that every pro-symmetric ring is symmetric. Several counterexamples are constructed to distinguish these classes of rings, and their mutual inclusion relations are also discussed.
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BibTeX
@misc{chen2026transposed,
title = {{Transposed Triple Products and Pro-Symmetric Rings in $\ast$-Rings}},
author = {Huaxi Chen and Long Wang and Honglin Zou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20084},
eprint = {2609.20084},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20084},
}
RIS
TY - UNPB
TI - Transposed Triple Products and Pro-Symmetric Rings in ∗-Rings
AU - Huaxi Chen
AU - Long Wang
AU - Honglin Zou
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20084
UR - https://arxiv.org/abs/2609.20084
AB - This paper investigates properties concerning transposed triple products in rings. Motivated by Cline's formula, we characterize symmetric rings by means of group invertible elements and EP elements. We prove that a unital ring R is symmetric if and only if abc∈ R^♯ implies acb∈ R^♯ for all a,b,c∈ R. In particular, we give an answer to the problem posed in \cite[Problem 2.9]{MW1}. An example is provided to illustrate that for a symmetric ring R, abc=e does not generally yield acb=e. For ∗-rings, we introduce the notion of pro-symmetric rings: a ring R is pro-symmetric if abc∈ P(R) implies acb∈ P(R) for all a,b,c∈ R. We show that every pro-symmetric ring is symmetric. Several counterexamples are constructed to distinguish these classes of rings, and their mutual inclusion relations are also discussed.
ER -