$\mathrm{TR}$ of quasiregular semiperfect rings is even

TL;DR

This paper proves that the TR groups in odd degrees are zero for all quasiregular semiperfect rings, revealing a specific structural property of these rings in algebraic K-theory.

Contribution

It establishes the vanishing of odd-degree TR groups for quasiregular semiperfect rings, a new result in algebraic K-theory.

Findings

TR_{2i+1}(S)=0 for all i and quasiregular semiperfect rings

Odd-degree TR groups vanish in this class of rings

Provides insight into the structure of algebraic K-theory for these rings

Abstract

We show that $\mathrm{TR}_{2i+1}(S)=0$ for all $i\in \mathbb{N}$ and all $S$ quasiregular semiperfect.