Strongly Nil-*-Clean Rings

TL;DR

This paper characterizes strongly nil-*-clean rings, showing they are equivalent to rings where all idempotents are projections, the ring is periodic, and the quotient by its Jacobson radical is Boolean, with additional results on algebraic extensions.

Contribution

It provides a characterization of strongly nil-*-clean rings and explores their properties and extensions, connecting them with Boolean *-rings.

Findings

R is strongly nil-*-clean iff all idempotents are projections, R is periodic, and R/J(R) is Boolean.

The algebraic extension R[i] is strongly nil-*-clean iff R is strongly nil-*-clean and μη is nilpotent.

Relationships between Boolean *-rings and strongly nil-*-clean rings are established.

Abstract

A *-ring $R$ is called a strongly nil-*-clean ring if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this article, we show that $R$ is a strongly nil-*-clean ring if and only if every idempotent in $R$ is a projection, $R$ is periodic, and $R/J(R)$ is Boolean. For any commutative *-ring $R$, we prove that the algebraic extension $R[i]$ where $i^2=\mu i+\eta$ for some $\mu,\eta\in R$ is strongly nil-*-clean if and only if $R$ is strongly nil-*-clean and $\mu\eta$ is nilpotent. The relationships between Boolean *-rings and strongly nil-*-clean rings are also obtained.