A Generalization Of ZI-rings And Some Applications

In this paper, we introduce the concept of quasi ZI-rings and discuss some properties of quasi ZI-rings and their relationships with some special rings, and study the non-singularity and regularity of quasi ZI-rings.Full text is divided into there chapters.In the first chapter, we mainly introduce the concept of quasi ZI-rings and discuss the properties of quasi ZI-ringsIn the second chapter, we mainly discuss the relationships of quasi ZI-rings with 2-primal rings,weakly Armendariz rings and weakly semicommutative rings.We mainly obtain some conclusions as follows: (1) If R is a quasi ZI-ring then R is a 2-primal ring.(2) If R is a quasi ZI-ring then R is a weakly Armendariz ring. (3) If R is a quasi ZI-ring then R is a weakly semicommutative ring.In the last chapter, we mainly study the non-singularity and regu-larity of quasi ZI-rings.We mainly obtain some conclusions as follows:(1) Let R be a ring. The following conditions are equivalent:(a) R is a strongly regular ring;(b) R is a quasi ZI,regular ring;(c) R is a quasi ZI,left SF-ring;(d) R is a quasi ZI,ELT ring whose simple left R-modules are either p-injective or projective. (2) If R is a quasi ZI-ring and whose simple left R-modules are either p-injective or projective, then R is left and right non-singular.