On fusible rings

Abstract

A ring R is called left fusible if every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor, and R is called uniquely left fusible if for any $a\in R$ there exists a unique left zero-divisor z such that a – z is non-left zero-divisor. We show that a left fusible ring R is uniquely left fusible if and only if either R is a domain or R has a unique non-left zero-divisor element.

Keywords:

• Fusible ring
• regular element
• unit
• zero-divisor

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D40
• 16D50
• 16D60
• 16S34

Acknowledgments

The authors would like to thank the referee for reading the paper carefully and giving many useful comments to clarify the whole paper.