Internal Cancellation over SSP Rings

TL;DR

This paper explores the properties of SSP rings with internal cancellation, showing that regular elements are special clean and establishing a connection between idempotent stable range 1 and internal cancellation.

Contribution

It introduces the concept that in SSP rings with internal cancellation, regular elements are special clean, and links internal cancellation with idempotent stable range 1.

Findings

Regular elements in SSP rings with internal cancellation are special clean.

Internal cancellation and idempotent stable range 1 coincide in SSP rings.

Characterization of internal cancellation over SSP rings by special clean elements.

Abstract

A ring is SSP if the sum of two direct summands is a direct summand. A ring has internal cancellation if every its (von Neumann) regular elements are unit-regular. We show that in an SSP ring having internal cancellation, any regular element is special clean. Our main results also imply that for any SSP ring internal cancellation and idempotent stable range $1$ coincide with each other. Internal cancellation over SSP was then characterized by special clean elements.