Internal Cancellation over SSP Rings

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.