Cofinitely Generalized Lifting Module

Let R be an associative ring with identity and let M be a unitary left R-module. In this paper, we introduce a ⊕-Jacobson-supplemented module (for short ⊕-J-supplemented), as a generalization of J-supplemented module. We state the main properties of a ⊕-J-supplemented modules and supplying by examples and remarks for these module and several properties of this modules are given. Also we introduce weakly ⊕-J-supplemented modules and give a characterization of weakly ⊕-J-supplemented modules a conditions under which the direct sum of weakly ⊕-J-supplemented modules is weakly ⊕-J-supplemented are given. We define a cofinitely ⊕-J-supplemented modules (for short, cof-⊕-J-supplemented) and some type of modules that are related to cof-⊕-J-supplemented module and prove properties of this type of modules. Also we discuss the relation between them with examples and remarks.

Let R be an associative ring with a unit and N be a left R-module. The set MR(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring ER(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if MR(N) = ER(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.

Continuing an earlier examination of the codominant dimension of rings and modules, a categorical characterization is given for rings of equivalent dominant and codominant dimensions. Specifically, the question is reduced to when R R and the left minimal injective cogenerator R can be used as test modules respectively for the dominant dimension of the projective modules and the codominant dimension of the injective modules. These conditions are in turn characterized by when the injective projective modules are Z-injective. Also, a new and shortened version is given for the proof that the codominant dimension of the injective modules is equal to the dominant dimension of the projective modules. Throughout, R is an associative ring with unity and all modules considered are unital. We denote a minimal injective cogenerator for R)R by R such being isomorphic to an injective hull of a direct sum of a complete representative class of simple left R-modules. For a left perfect ring R and a left R-module RM we define the codominant dimension of RM, denoted codom dim RM, to be greater than or equal to n if for a minimal projective resolution . P* P M 0, Pi is injective for i = 1, 29, . . ., n. The left codominant dimesnion of R is defined to be codom dim RU. The dominant dimension of RM is defined in a dual manner, and the left dominant dimension of R is dom dimRR. Further, for R left perfect, we define codom dimR9 to be infIcodom dim RE: RE injectivel. Similarly, dom dim Ris given by inf dom dimRP: RP projectivel. Finally, we say that RE (nRP) is a test module for RI (RT) provided codom dim RE= codom dimRI (dom dimR P = dom dimRY). In [2] we showed that dom dim R 9= codom dim RI. As an application it can Presented to the Society, January 25, 1973; received by the editors April 16, 1973. AMS (MOS) subject classifications (1970). Primary 16A60; Secondary 16A50, 16A52.

Let R be an associative ring with unit element. Mod-R and R-Mod will denote the categories of unitary right and left R-modules, respectively, and all modules are assumed to be in Mod-R unless otherwise specified. For all M, N ϵ Mod-R, HomR(M, N) will usually be abbreviated as [M, N]. For the definitions of basic terms, and an exposition on torsion theories in Mod-R, the reader is referred to Lambek [6]. Jans [5] has called a class of modules which is closed under submodules, direct products, homomorphic images, group extensions, and isomorphic images a TTF (torsion-torsionfree) class.

Let R be an associative ring which is not necessarily commutative. For any torsion theory τ on the category of left R-modules and for any nonnegative integer n we define and study the notion of the nth local cohomology functor with respect to τ. For suitably nice rings a bound for the nonvanishing of these functors is given in terms of the τ-dimension of the modules.

Let R be an associative ring with identity, and let denote the category of unital left R-modules. The Walkers [6] raised the question of characterizing the maximal torsion radicals of , and showed that if R is commutative and Noetherian, then there is a one-to-one correspondence between maximal torsion radicals and minimal prime ideals of R [6, Theorem 1.29]. Popescu announced [5, Theorem 2.5] that the result remains valid for commutative rings with Gabriel dimension (in the terminology of [2]). Theorem 4.6 below shows that the result holds for rings (not necessarily commutative) with Krull dimension on either the left or right, extending the previous theorem for right Noetherian rings which appeared in [1].

Throughout this paper, R wilt denote an associative ring with unity, all the R-modu!es RA = A will be left unitat R-modules, and all mappings f : X-+ Y between R-modules will be R-homomorphisms, unless otherwise specified. We shall be concerned with an arbitrary, but fixed class F of R-monomorphisms e: A--~ B. If C.--Coker :~, then we shall say that the arising exact sequence E: 0 . -~A-~ B-~ C--~ 0 is a F-sequence. d will denote the class of the domains of the maps of F, ~ the class of codomains, and ~ the class of the cokernels. An R-module M will be called injective with respect to ~ or E4)~jective if for each homomorphism f : A --=~ M, there exists an f : B-+ M such that f.~, ~ = f We define an R-modute M to be F-injective if M is injective with respect to ~z, ~or each ~sF. This paper wilt examine F-injecti~Jty in general, establishing its properties and pointing out how these properties generalize know.t~ results for previously studied types of injectivity and divisibility~ Examples includfng two new ones are discussed in tI. Sections III and IV are devoted to the question as to when the class of F-injective modules is closed with respect to epimorphic images. Section V gives a necessary and sufficient condition that all R-modules contain maximum F-injective submodules. In VI and VII, respectively~ new concepts of purity and torsionfreeness are defined and studied in relation to F-injectivity,

We will add new types of modules over a ring ℜ called generalized semi-regular modules. Assuming ℜ is a ring with identity and M is a unitary left ℜ-module which is a duality. We show that our new ideas have characterization and a few properties.