A New Characterization of Semi-Tripotent Rings
Ahmad Moussavi, Peter Danchev, Arash Javan, Omid Hasanzadeh
TL;DR
Problem: characterize semi-tripotent rings and obtain a new characterization via the DT framework built on $\\Delta(R)$ and $\\operatorname{Tr}(R)$, including criteria for when group rings are semi-tripotent. Method: develop the $\\Delta$-tripotent (DT) theory, prove DT is equivalent to semi-tripotent rings, analyze $RG$ through the torsion structure of the group $G$ and the primes in $\\Delta(R)$, and decompose $R/ J(R)$ into Boolean and Yaqub components. Contributions: (i) a new element-based DT characterization of semi-tripotent rings; (ii) precise group-ring results forcing $G$ to be a torsion $p$-group with $p\\in\\Delta(R)$ and detailed restrictions for $2$- or $3$-groups; (iii) a structural decomposition $R/ J(R)\\cong R_1\\times R_2$ with $R_1$ Boolean/DI and $R_2$ Yaqub/semi-tripotent. Significance: extends Ko\\u{s}an et al.'s work, provides concrete tests and constructions for semi-tripotent behavior in group rings and DT rings, and equips researchers with practical criteria to identify semi-tripotent structure in ring-theoretic contexts.
Abstract
We give a comprehensive study of the so-called \textit{semi-tripotent rings} obtaining their new and non-trivial characterization as well as a complete description in terms of sums and products of some special elements. Particularly, we explore in-depth when a group ring is semi-tripotent. Our results somewhat supply those established by Ko$ş$an et al. in Can. Math. Bull. (2019).