K₀ Groups Of Some Rings, REFINEMENT Rings And Comparability

In this dissertation, we consider the Grothendieck group of abelian exchange rings, and introduce the notions of refinement rings and modules and investigate their prop-erties. We also study the some comparability of refinement rings, the relations of com-parability of regular rings and excellent extensions. Finally, we consider the strongly 7r-regular rings without unit.In Chapter 2, let R be a ring, and let S(R) be the nonempty set of all the proper ideals of R generated by central idempotents. Recall that if P is a maximal ele-ment of the set S(R), then the factor ring R/P is called a Pierce stalk of R. Let (?)(R)={P E S(R)│R/P is a pierce stalk of R}. We first point out that (?)(R) is a compact space with Zariski topology, and then we shall give a characterization of the Grothendieck group of an abelian exchange ring, i.e., there is an order-preserving isomorphism (K0(R), [R])≌((?)(R), u), where (?)(R)={f:(?)(R)→Z│f is contin-uous}.In Chapter 3, we show that the finitely generated left R-module M is refinement iff S=EndR(M) is a left refinement ring. Moreover, we study the separativity of refinement modules (rings).It is well-known that any exchange ring R has the following properties:(1) let I be an ideal of R, for any M'∈FP(R/I), there is M E FP(R) such that R/I (?)R M≌M', which we call finitely generated projective left R/I-module M'can be lifted to M E FP(R);(2) whenever A, B E FP(R) with A/IA≌B/IB, there exist decompositions A=A1(?)A2 and B=B1(?)B2 such that A1≌B1, A2=IA2, B2=IB2.There are other rings without exchange property which have the properties (1) and (2). Let R be a refinement ring and I be a nonzero ideal of R. We shall prove that if R has the above properties, then R/I and I are refinement rings. In particular, if I=J(R) is left T-nilpotent and every finitely generated projective left R/J(R)-module can be lifted, then R/J(R) is a left refinement ring if and only if R is a left refinement ring where J(R) is the Jacobson radical of R. Let R be a left refinement ring and I be a nonzero ideal of R. If R satisfies the above properties, then R is separative iff so are I and R/I.In Chapter 4, we study the comparability of refinement rings. The comparability concepts have been proved to be particularly fruitful in the development of the theory of regular rings. Let R be a refinement ring. If every nonzero finitely generated projective R-module has a nonzero cyclic submodule as its direct summand, then the following conditions are equivalent:(1) R satisfies almost comparability;(2) for each finitely generated projective R-module P, EndR(P) satisfies almost comparability;(3) every ring S which is Morita equivalent to R satisfies almost comparability;(4) for all positive integers n, Mn(R) satisfies almost comparability;(5) there exists a positive integer n such that Mn(R) satisfies almost comparability.Let S be an excellent extension of a (von Neumann) regular ring R. In Chapter 5, we study comparability of S related to comparability of R. If R has the n-unperforation property, then R satisfies s-comparability (almost comparability or weak comparabil-ity) if and only if so does S.In Chapter 6, we say that a ring without unit I is a strongly 7r-regular ring if for any x∈I, there exist y∈I, n∈N such that xn=xn+1y. We give some characterizations of the strongly 7r-regular ring without unit. Let I be an abelian ring without unit. Then I is a stronglyπ-regular ring if and only if N(I)=J(R)∩I and I/N(I) is regular, where R is the unitization of I, and N(I) is the set of all of nilpotent elements of I. Let (I, J, M, N,φ,Ψ) be Morita context with zero pairings, and let T be the ring of Morita context of it. Then T is a strongly 7r-regular ring of bounded index if and only if so are I and J.