Small pseudo projective modules, small pseudo stable submodule 1 introduction throughout this paper the basic ring r is supposed to be ring with unity and all modules are unitary left rmodules. Likewise, projective and flat resolutions are left resolutions such that all the e i are projective and flat r modules, respectively. Projective and injective modules arise quite abundantly in nature. M to indicate that n is an essential submodule, a small. Introductory examples of flowcharts and pseudocode chapter 3 calculate pay sequence start end input hours print pay pay hours rate input rate begin input hours input rate pay hours rate print pay end 2 sum of 2 numbers sequence start end input x input y output sum. Introduction the aim of this note is to construct left r modules m, where ris a kalgebra over some eld k, with the following properties. Recall that a submodule n of a module a is called small pseudo stable, if for any. 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. In addition, a splitting property for projective modules recently established by gabber, liu and lorenzini is also discussed.
A right rmodule n is called pseudo principally minjective if every rmonomorphism from a principal submodule of m to n can be extended to an homomorphism fror m m to n. Projective modules with finitely many generators are studied in algebraic theory. Indecomposability of m and the localness of end a m. An module is projective if and only if there exists an module such that is free proof. Iwasawa theory, projective modules, and modular representations. An module is projective if and only if there exists an module such that is free. A short exact sequence of amodules is a sequence of the form 0. Oct 28, 2011 throughout is a ring with 1 and all modules are left modules.
Semisimple modules fix a ring rnot necessarily commutative. Talebi and others published on pseudo projective and pseudosmallprojective modules find, read and cite all the research you need on researchgate. We have provide an example of module which are semiprojective but not quasiprojective. A sufficient condition for a small pseudo projective module to be a s. For example, in most treatments i know the the definition of projective module is given as either. In particular p is pseudo nite provided it is projective in. Perhaps the most successful invariant measuring their complexity is the invariant. Pseudo projective module, pseudo weakly projective module, projective cover. Pdf on semiprojective modules and their endomorphism rings. If trm is finitely generated, as remarked earlier, it can be generated by an idempotent trm re and m can be viewed as a projective module over re with unit trace. Chinese remainder theorem, endomorphisms of projective modules, projective but not free, tensor product of projective modules 2 so far we have only given a trivial example of projective modules, i. Projective modules are important for at least the following reasons. The following fact shows how to convert addm into a category of projective modules.
Hypergeometric systems and projective modules in hypertoric category o we prove that indecomposable projective and tilting modules in hypertoric category oare obtained by applying the geometric jacquet functor of emerton, nadler, and vilonen to certain gelfandkapranovzelevinsky hypergeometric systems. Notice that z2z and z3z are z6zmodules and we have an isomorphism of z6zmodules. Note that mis projective i for all exact sequences 0. Pseudoprojective modules quasiprojective modules amodulem is called an automorphisminvariant module if every isomorphism between two essential submodules of m extends to an automorphism of m. You might also want to take a look at the article when every projective module is a direct sum of finitely generated modules by w. The solution appears to rely on the reasoning of, but it isnt clear why one could reason with dimensions this way. The difference between free and projective modules is, in a sense, measured by the algebraic ktheory group k 0 r, see below. In particular one gets very easy but not very satisfying examples by looking at disconnected rings. In previous lecture notes, one can nd the theory of injective and projective modules.
In analogy with the terminology local in the commutative case, the algebra a is called local if a. For example, a free resolution of a module m is a left resolution in which all the modules e i are free rmodules. Some results on pseudo semiprojective modules in this section, we study some properties of pseudo semiprojective module and its endomorphism ring. Then m is finitely generated iff trm is finitely generated.
Irving kaplansky, projective modules, the annals of mathematics second series, vol. The reason this seems simple is that there are many equivalent definitions of projective module, and what you give as the definition is usually a property that is shown to be equivalent. Projective modules over dedekind domains, february 14, 2010. We do not expect that every pseudo nite module is pself.
The study of the relation between x and the topology of x and its. On pseudoprojective and pseudosmallprojective modules hikari. But then f j f j for every j2j, and since this equality holds for every generator f j of f, it is easy to deduce that, as required. Wecallamodule m tobeadualautomorphisminvariant module if whenever k1 and k2 are small submodules. Some results on pseudo semi projective modules in this section, we study some properties of pseudo semi projective module and its endomorphism ring.
On ding injective, ding projective and ding flat modules and complexes gillespie, james, rocky mountain journal of mathematics, 2017. Throughout is a ring with 1 and all modules are left modules corollary 1. Cinjectivity and cprojectivity yue chi ming, roger, hiroshima mathematical journal, 2007. Then m is called projective if for all surjections p. Projective and injective modules hw pushouts and pullbacks. If is free, then is projective by lemma 2 in part 1. Pseudo projective modules which are not quasi projective and quivers deste, gabriella and tutuncu, derya keskin, taiwanese journal of mathematics, 2018 euclid. It is also provided the sufficient condition for pseudo mp.
The \only if follows from the same argument as the easy direction of. Symplectic topology of projective manifolds with small. Cosemisimple modules and generalized injectivity liu, zhongkui and ahsan, javed, taiwanese journal of mathematics, 1999. It is a long held belief in physics that the notion of spacetime as a pseudo riemannian manifold requires modi. Likewise, projective and flat resolutions are left resolutions such that all the e i are projective and flat rmodules, respectively. In this thesis, we study the theory of projective and injective modules. Projective modules are recognized by having trivial invariant, 2, 3. Any pseudoprojective module is dual automorphisminvariant. 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 have studied the properties of semiprojective module and its endomorphism rings related with hopfian, cohopfian, and directly finite modules. There are rings such that as modules, and so they do not have a unique dimensionrank.
An algebraic formulation is given for the embedded noncommutative spaces over the moyal algebra developed in a geometric framework in 8. This paper introduces the notion of dual ofsuchmodules. The coincidence of the class of projective modules and that of free modules has been proved for. Throughout this paper all rings are associative rings with identity, and all. Clearly every selfsmall module is pseudo nite, and it is easy to see that any direct summand of a direct sum of nitely generated modules is pseudo nite. On pseudo semiprojective modules academic journals. Let m be an rmodule, a submodule k of m is said to be.
We explicitly construct the projective modules corresponding to the tangent bundles of the. Pseudo principally quasiinjective modules and pseudo. First of all, the structure of quasiprincipally projective modules was. Endomorphism rings of small pseudo projective modules. Pseudo projective modules quasi projective modules amodulem is called an automorphisminvariant module if every isomorphism between two essential submodules of m extends to an automorphism of m. A short exact sequence of a modules is a sequence of the form 0. Thus, we will call manifolds with small dual also manifolds with positive defect. For example, all free modules that we know of, are projective modules. Now we localize at a maximal ideal m and since this is. However, in the case of a finite dimensional algebra like this matrix ring, its easy to argue via dimensions that the rank is unique.
If ris a pid then every nitely generated projective rmodule is free. Algebra, 44 2016, 51635178, sci phan the hai, truong cong quynh and le van thuyet, mutually essentially pseudo injective modules, the bulletin of the malaysian mathematical sciences society, 39 2 2016, 795803, scie. Over rings decomposable into a direct sum there always exist projective modules different from free ones. It is proved that a pseudo mpprojective module is hop. For example, a free resolution of a module m is a left resolution in which all the modules e i are free r modules. In this paper characterization of pseudo mpprojective modules and quasi pseudo principally projective modules are given and discussed the various properties of it. International journal of computer applications 0975 8887. In this paper characterization of pseudo mp projective modules and quasi pseudo principally projective modules are given and discussed the various properties of it. See the history of this page for a list of all contributions to it.
A module mr is called pseudo fqinjective or pfqinjective for short if every monomorphism from a finitely generated submodule of m to m. The simplest example of a projective module is a free module. Recall that a ring r is called quasifrobeinus if it is left. Z2z z3z thus z2z and z3z are nonfree modules isomorphic to direct summands of the free. In other words, projective modules are the way to express vector bundles in algebraic language. Various equivalent characterizations of these modules appear below. Note that this is not an intrinsic property of x, but rather of a given projective embedding of x the class of algebraic manifolds with small dual was studied by many authors, for instance, see 5, 20, 21, 27, 31, 45, see also for a survey. Let m be a small pseudo projective module then m is s. It is proved that a pseudo mpprojective module is hopfian iff mn is hopfian, for each fully invariant small submodule n of m. Let m be an r module, a submodule k of m is said to be.
Example of a projective module which is not a direct sum. Then there is a natural equivalence between addm and the category of projective right smodules. Bbe faithfully at, and ma nite amodule anoetherian. By remark 2 in part 1, there exists an exact sequence where is free. Every free module is a projective module, but the converse fails to hold over some rings, such as dedekind rings. In this paper, we introduce the concept of esmall mprojective modules as a. In this research, we introduce the definition of pseudo principally quasiinjective modules and pseudo quasiprincipally injective modules and give characterizations and properties of these modules which are extended from the previous works. It is proved that a pseudo mp projective module is hop. In this paper we construct pseudo projective modules which are not quasi projective over noncommutative perfect rings. Semiprojective module, pseudo principally projective. Projective modules and embedded noncommutative spaces 5 given an integer m n, we let lam. On pseudoprojective and pseudosmallprojective modules.
On image summand coinvariant modules and kernel summand. Section 3, we investigate radical projective modules. These modules had to do with splitting of certain exact sequences. In many circumstances conditions are imposed on the modules e i resolving the given module m. Then m is projective if and only if m b is a projective as a bmodule, of course.
Small pseudo projective modules, small pseudo stable sub module 1 introduction throughout this paper the basic ring r is supposed to be ring with unity and all modules are unitary left r modules. It is proved that a pseudo mp projective module is hopfian iff mn is hopfian, for each fully invariant small submodule n of m. We first give some fundamental properties of imsummand coinvariant modules and imsmall coinvariant modules, and we prove that, for modules m and n over a right perfect ring such that n is a small epimorphic image of m, m. If ris a pid, fis a free rmodule of a nite rank, and m f is a submodule then mis a free module and rankm rankf. Introduction the aim of this note is to construct left rmodules m, where ris a kalgebra over some eld k, with the following properties. In this paper we generalize the basic properties of pseudo weakly projective modules.
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