29 research outputs found
ON GENERALIZED EPI-PROJECTIVE MODULES
A module M is said to be generalized N-projective (or N-dual ojective) if, for any epimorphism g : N → X and any homomorphism f : M → X, there exist decompositions M = M1 ⊕ M2, N = N1 ⊕ N2, a homomorphism h1 : M1 → N1 and an epimorphism h2 : N2 → M2 such that g ◦ h1 = f|M1 and f ◦ h2 = g|N2 . This relative projectivity is very useful for the study on direct sums of lifting modules (cf. [5], [7]). In the definition, it should be noted that we may often consider the case when f to be an epimorphism. By this reason, in this paper we define relative (strongly) generalized epi-projective modules and show several results on this generalized epi-projectivity. We apply our results to the known problem when finite direct sums M1⊕· · ·⊕Mn of lifting modules Mi (i = 1, · · · , n) is lifting.</p
ON MONO-INJECTIVE MODULES AND MONO-OJECTIVE MODULES
In [5] and [6], we have introduced a couple of relative generalized
epi-projectivities and given several properties of these projectivities.
In this paper, we consider relative generalized injectivities that are
dual to these relative projectivities and apply them to the study of direct
sums of extending modules. Firstly we prove that for an extending
module N, a module M is N-injective if and only if M is mono-Ninjective
and essentially N-injective. Then we define a mono-ojectivity
that plays an important role in the study of direct sums of extending
modules. The structure of (mono-)ojectivity is complicated and hence it
is difficult to determine whether these injectivities are inherited by finite
direct sums and direct summands even in the case where each module
is quasi-continuous. Finally we give several characterizations of these
injectivities and find necessary and sufficient conditions for the direct
sums of extending modules to be extending
On some radicals and proper classes associated to simple modules
For a unitary right module , there are two known partitions of simple modules in the category : the first one divides them into -injective modules and -small modules, while the second one divides them into -projective modules and -singular modules. We study inclusions between the first two and the last two classes of simple modules in terms of some associated radicals and proper classes
Absolute co-supplement and absolute co-coclosed modules
A module M is called an absolute co-coclosed (absolute co-supplement) module if whenever M ≅ T/X the submodule X of T is a coclosed (supplement) submodule of T. Rings for which all modules are absolute co-coclosed (absolute co-supplement) are precisely determined. We also investigate the rings whose (finitely generated) absolute co-supplement modules are projective. We show that a commutative domain R is a Dedekind domain if and only if every submodule of an absolute co-supplement R-module is absolute co-supplement. We also prove that the class Coclosed of all short exact sequences 0→A→B→C→0 such that A is a coclosed submodule of B is a proper class and every extension of an absolute co-coclosed module by an absolute co-coclosed module is absolute co-coclosed.Scientific and Technical Research Council of Turke
Some rings for which the cosingular submodule of every module is a direct summand
The submodule Z(M) = ∩{N | M/N is small in its injective hull} was introduced by Talebi and Vanaja
in 2002. A ring R is said to have property (P ) if Z(M) is a direct summand of M for every R-module M . It is
shown that a commutative perfect ring R has (P ) if and only if R is semisimple. An example is given to show that
this characterization is not true for noncommutative rings. We prove that if R is a commutative ring such that the class
{M ∈ Mod−R | ZR(M) = 0} is closed under factor modules, then R has (P ) if and only if the ring R is von Neumann
regular