Remark on Bi-Ideals and Quasi-Ideals of Variants of Regular Rings

Problem statement: Every quasi-ideal of a ring is a bi-ideal. In general, a bi-ideal of a ring need not be a quasi-ideal. Every bi-ideal of regular rings is a quasi-ideal, so bi-ideals and quasi-ideals of regular rings coincide. It is known that variants of a regular ring need not be regular. The aim of this study is to study bi-ideals and quasi-ideals of variants of regular rings. Approach: The technique of the proof of main theorem use the properties of regular rings and bi-ideals. Results: It shows that every bi-ideal of variants of regular rings is a quasi-ideal. Conclusion: Although the variant of regular rings need not be regular but every bi-ideal of variants of regular rings is a quasi-ideal.