Pdf this paper provides the several homological characterization of perfect rings and semisimple rings in terms of semiprojective modules. Projective and injective modules play a crucial role in the study of the cohomology of representations. Journal of pure and applied algebra vol 224, issue 6. Good tilting modules and recollements of derived module categories, ii chen, hongxing and xi, changchang, journal of the mathematical society of japan, 2019. Pseudo projective modules which are not quasi projective and quivers deste, gabriella and tutuncu, derya keskin, taiwanese journal of mathematics, 2018. Generalization of semiprojective modules article pdf available in international journal of computer applications 838. Rings over which every module has a flat \\deltacover pinar aydogdu turk j math, 37, 20, 182194. Suppose first that is projective and let by corollary 1, there exists an module and a free module such that but then and thus is projective by corollary 1 again. Let m be a small pseudo projective module then m is s. Over rings decomposable into a direct sum there always exist projective modules different from free ones. If f is a free rmodule and p f is a submodule then p need not be free even if pis a direct summand of f. A right rmodule m is called semiprojective if, for any submodule n of m, every epimorphism.
Let m be any pseudo projective module and endm denotes the endomorphism ring of m. Properties of semiprojective modules and their endomorphism. A ring r is called semi hereditary if every finitely generated right ideal. Then addm will denote the full subcategory of the category of right r modules whose objects are direct. Talebi and others published on pseudo projective and pseudo small projective modules find, read and cite all the research you need on researchgate. Vasconcelos1 one of the aims of this paper is to answer the following question. T h e n, ker f is a dire ct sum mand of m if and only if f m is.
Mr3006737 indexed quynh, truong cong on pseudo semiprojective modules. The coincidence of the class of projective modules and that of free modules has been proved for. The modules which are constructed will have finite length three and only three nonzero proper submodules. Endomorphism rings of small pseudo projective modules 605 proposition 4. Jul 21, 2016 let g be a semi simple algebraic group over a perfect field k. Some results on pseudo semi projective modules in this section, we study some properties of pseudo semi projective module and its endomorphism ring. We have provide an example of module which are semi projective but not quasi projective. Let a be a commutative ring for which projective ideals are finitely generated. Xp1 is pseudoisomorphic to a direct sum of modules of the form ea. Similarly, the group of all rational numbers and any vector space over any eld are examples of injective modules.
A sufficient condition for a small pseudo projective module to be a s. In this paper characterization of pseudo mp projective modules and quasi pseudo principally projective modules are given and discussed the various properties of it. Pseudo projective module, pseudo weakly projective module, projective cover. Ct 15 mar 2019 pseudodualizing complexes and pseudoderived categories leonid positselski abstract. It is easy to show that if mis quasi pseudo principally projective, then every epimorphism in endm ris an automorphism. Pseudo projective and essentially pseudo injective modules 419 proposition 7. Endomorphism rings of small pseudo projective modules. By module we will mean rmodule and by homomorphism respectively isomorphism we will mean homomorphism respectively isomorphism of rmodules, unless explicitly stated otherwise. Projective modules over dedekind domains, february 14, 2010. Let n and m be mutually pseudo projective modules and a be any submodule of n such that na is isomorphic to a direct summand of m,then a is a direct summand of n. A lot of progress has been made recently in computing the chow motives of projective ghomogeneous varieties.
Complexes in this paper ris a ring, and the default action on modules is on the left. In this thesis, we study the theory of projective and injective modules. For example, all free modules that we know of, are projective modules. Chinese remainder theorem, endomorphisms of projective modules, projective but not free, tensor product of projective modules. Oct 28, 2011 examples of projective modules posted. Then is projective if and only if is projective for all. Prakasha and others have extended this notion to kenmotsu and lp. Projective and injective modules arise quite abundantly in nature. The simplest example of a projective module is a free module. Dec 23, 2016 the question is addressed of when all pure. Alternatively, another fast way to prove by some theorems is that any modules over a pid, it is a projective module if and only if it is a free module.
On pseudoprojective and pseudosmallprojective modules 1. An algebraic formulation is given for the embedded noncommutative spaces over the moyal algebra developed in a geometric framework in 8. It is wellknown that the submodules of a projective module need not be projective. On aquasiprojective modules and asemiperfect modules. This part ii is devoted to a detailed study of all the 3dimensional local. The main reasons that i am choosing this particular topic in noncommutative algebra is for the study of representations of nite groups which we will do after the break. On pseudoprojective and pseudosmallprojective modules. The difference between free and projective modules is, in a sense, measured by the algebraic ktheory group k 0 r, see below. Throughout this paper all rings are associative rings with identity, and all. In part i, we have exhibited a 6dimensional kalgebra. If every module has a pseudo projective cover then it has a projective cover.
When k has positive characteristic, a broader class of ghomogeneous varieties appear. We explicitly construct the projective modules corresponding to the tangent bundles of the. Every semi simple projective module is pseudo projective module. In this paper, the concept of quasi pseudo principally injective modules is introduced and a characterization of commutative semi simple rings is given in terms of quasi pseudo principally injective modules. Iwasawa theory, projective modules, and modular representations. Assuming the axiom of choice, then by the basis theorem every module over a field is a free module and hence in particular every module over a field is a projective module by prop. In this paper we construct pseudo projective modules which are not quasi projective over noncommutative perfect rings. Nov 19, 2016 in this paper, we have studied the properties of semi projective module and its endomorphism rings related with hopfian, cohopfian, and directly finite modules. Furthermore, one of the authors has given some characterization of semi perfect modules, defined in. Pdf this paper provides the several homological characterization of perfect rings and semisimple rings in terms of semi projective modules.
Quasipseudo principally injective modules algebra colloquium. These are varieties over which g acts transitively with possibly nonreduced isotropy subgroups. Recall that a ring r is called quasifrobeinus if it is left. Projective modules over dedekind domains, february 14, 2010 3 2. Some results on pseudo semiprojective modules in this section, we study some properties of pseudo semi projective module and its endomorphism ring.
An example of pseudo mpinjective module which is not m pseudo injective is given. International journal of computer applications 0975 8887. Quynh have studied the same under the name pseudo semiprojective modules, now authors are in position to prove the various property of such modules. Thus, r modules are left r modules, while right r modules are considered to be left modules over the opposite ring r. On categories of projective modules manabu harada and hikoji kanbara received march 2, 1971 the authors have studied some structures in categories of completely indecomposable modules in 5, 6 and 7, respectively. Pdf in this paper characterization of pseudo mpprojective modules and quasi pseudo principally projective modules are given and discussed the. Asianeuropean journal of mathematics vol 11, no 02. It is proved that a pseudo mpprojective module is hopfian iff mn is hopfian, for each fully invariant small submodule n of m. It is proved that this is the case over hereditary noetherian rings. Pdf on semiprojective modules and their endomorphism rings.
In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. A semisimple rmodule is a nite direct sum of simple modules m s 1 s n and a semisimple ring is a ring rfor which all f. We will write pim for a projective indecomposable amodule. In this paper, we consider some generalizations of semi projective module, that is quasi pseudo principally projective module. So by contradiction, suppose q is a projective z module. This paper shows that properties of projective modules over a group ring zp.
A module is said to be projective if it is a direct summand of a free module. Projective modules with finitely many generators are studied in algebraic theory. Projective modules over local rings before we can consider projective modules over dedekind domains, we will consider the case of projective modules over noetherian local rings. Ct 15 mar 2019 pseudo dualizing complexes and pseudo derived categories leonid positselski abstract. Pdf generalization of semiprojective modules researchgate. In this paper we generalize the basic properties of pseudo weakly projective modules. This tensor field p is known as pseudo projective curvature tensor. Modules which are epiequivalent to projective modules. On pseudoprojective and essentially pseudo injective modules. Pdf projective modules with semilocal endomorphism ring. Show full abstract some properties of this classes modules are. In this paper characterization of pseudo mpprojective modules and quasi pseudo principally projective modules are given and discussed the various properties. In this article we investigate a certain projectivity which is preserved under the taking of submodules.