Weakly Euclidean Rings And Modules

In the first chapter, we mainly introduce the background knowledge of Euclidean rings, the further research about it and main results of the thesis.In the second chapter, we review some basic concepts and properties such as rings and modules, ideals, rings of quotient, cyclic modules, which is classical in abstract algebra, and then introduce some known results about Euclidean domains.We mainly investigate matrix structures overω-Euclidean rings in the third chap-ter. The chapter has three parts. In the first part, we introduce the basic properties ofω-Euclidean rings. In the second part, we consider the problems of elementary re-duction of matrices overω-Euclidean rings, and prove that a rightω-Euclidean ring is a right Hermite ring; all finitely generated ideals are principal; any invertible matrix overω-Euclidean rings is a finite product of elementary matrices; Also we investigate some conditions of elementary reduction of matrices overω-Euclidean rings. In the third part, we will further explore the stable rank overω-Euclidean rings.In the fourth chapter, we will defineω-Euclidean modules and obtain some ele-mentary properties. We show that the submodules ofω-Euclidean modules are cyclic, and prove that the homomorphic kernels and the homomorphic images ofω-Euclidean modules are againω-Euclidean. We also discuss the existence of the great common divisors forω-Euclidean modules. Afterwards, we mainly discussω-Euclidean rings and endomorphism rings ofω-Euclidean modules, and analyseω-Euclidean rings by virtue of cyclic modules. We show that a commutative ring R is anω-Euclidean ring if and only if for every cyclic R-module M, End(RM)is anω-Euclidean ring.The fifth chapter introduce the concept ofω-Euclidean ideals, and define Eω-injective modules,Eω-projective modules and Eω-flat modules. We consider them from the homological viewpoints:the direct sum of Eω-projective modules is still Eωprojective; the direct product and direct sum of Eω-injective modules are Eω-injective; the direct sum of Eω-flat modules is still Eω-flat; there exists a equivalent characteri-zation between Eω-injective modules and Eω-projective modules. These results lay a foundation for further research onω-Euclidean rings without identity.