On the upper nilradical and Jacobson radical of rings graded by t.u.p.-semigroups

TL;DR

This paper proves that in rings graded by t.u.p.-semigroups, both the nilradical and Jacobson radical are homogeneous, addressing open questions in the structure theory of such rings.

Contribution

It establishes that the nilradical and Jacobson radical are homogeneous in rings graded by t.u.p.-semigroups, providing partial answers to existing open problems.

Findings

Nilradical is homogeneous in t.u.p.-semigroup graded rings.

Jacobson radical is homogeneous in t.u.p.-semigroup graded rings.

Addresses open questions by Smoktunowicz and Jespers.

Abstract

Given a t.u.p.-semigroup (two unique product semigroup) $X$, we show if $R$ is an $X$-graded ring then both its nilradical and Jacobson radical are homogeneous. This partially answers questions of Smoktunowicz and Jespers.