Euclidean Rings And Their Generalization

In the first chapter, we generalize the notion of Euclidean domain which is classical in abstract algebra, and define weakly Euclidean rings and Euclidean modules.In the first section we mainly discuss the basic properties of Euclidean modules. We can see that submodules of Euclidean modules are cyclic, but there are cyclic modules which are not Euclidean and we give a counterexample. We also prove that the submodules and homomorphic images of Euclidean modules are again Euclidean, but its inverse may be wrong and we can give a counterexample too.In the second section, we mainly discuss weakly Euclidean rings and endomorphism rings of Euclidean modules. We prove that a commutative ring R is weakly Euclidean if and only if for every cyclic R-module M, End(_RM) is a weakly Euclidean ring. As a corollary of this theorem, we obtain an interesting characterization of left global dimension of left weakly Euclidean rings. If R is a left weakly Euclidean ring, then LgdR=Sup{PdN| N is a left Euclidean module }.In the third section we mainly deal with torsion-free Euclidean modules. We can see many good properties of torsion-free Euclidean modules. We prove that torsion-free Euclidean R-modules must be projective modules, uniform modules, Dedekind modules and multiplication modules.The second chapter defines E—ideals, and we can easily see that they are contained in principal ideals and contain minimal ideals. Then E-injective modules, E-projective modules, E-hereditary rings, E-flat mod- ules, E-regular rings and E-coherent rings are introduced. In section 1, we prove that E-injective modules are closed not only to direct product but also to direct sum. Similar to the situations of injective modules, we can also use the functor Ext to characterize E—injective modules. If R is a left E—hereditary rings, then every factor module of E-injective modules is again E-injective. In the final part of this section, we use E-injective modules to define the E-injective dimension of modules and E-global dimension of rings. Under this definition, the left E-global dimension of left E-hereditary rings≤1. In section 2, we use the functor Tor to characterize E-flat modules and prove that R-module M is E-flat if and only if the characteristic module M* is E-injective. Then we demonstrate that pure submodules and direct sum of E-flat modules are again E-flat. We see that if R is left E-regular, then every left R-module is E-injective and every right R-module is E-flat. At the same time, many equivalent characterizations of E-coherent rings are given in this section. In section 3, we define E-injective rings and prove some properties. The main result of this section is: if R is both right weakly Euclidean and right E-injective, then J(R) = Z(R_R).