On Rings with the Goodearl–Menal Condition

Abstract

A ring R is said to satisfy the Goodearl–Menal condition if for any x, y ∈ R, there exists a unit u of R such that both x − u and y − u ⁻¹ are units of R. It is proved that if R is a semilocal ring or an exchange ring with primitive factors Artinian, then R satisfies the Goodearl–Menal condition if and only if no homomorphic image of R is isomorphic to either ℤ₂ or ℤ₃ or 𝕄₂(ℤ₂). These results correct two existing results. Some consequences are discussed.

Key Words:

• Goodearl–Menal condition
• Matrix ring
• Semilocal ring
• Unit
• Unit 1-stable range

2010 Mathematics Subject Classification:

• Primary 16S50
• 16U99

ACKNOWLEDGMENTS

The authors are grateful to the referee for his/her thorough reading the manuscript and for valuable comments and suggestions which have improved the presentation of this work. Chunna Li acknowledges support by the State Scholarship Fund from China Scholarship Council. Lu Wang and Yiqiang Zhou acknowledge support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

Notes

Communicated by I. Swanson.