Cosemisimple modules and generalized injectivity liu, zhongkui and ahsan, javed, taiwanese journal of mathematics, 1999. 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. Let m be an rmodule, a submodule k of m is said to be. In particular one gets very easy but not very satisfying examples by looking at disconnected rings. For example, a free resolution of a module m is a left resolution in which all the modules e i are free r modules. Similarly, the group of all rational numbers and any vector space over any eld are examples of injective 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. On pseudoprojective and pseudosmallprojective modules. A short exact sequence of a modules is a sequence of the form 0.
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. Oct 28, 2011 throughout is a ring with 1 and all modules are left modules. The difference between free and projective modules is, in a sense, measured by the algebraic ktheory group k 0 r, see below. Any pseudoprojective module is dual automorphisminvariant.
Some results on pseudo semiprojective modules in this section, we study some properties of pseudo semiprojective module and its endomorphism ring. Pseudo projective module, pseudo weakly projective module, projective cover. Thus, we will call manifolds with small dual also manifolds with positive defect. Let m be a small pseudo projective module then m is s.
Indecomposability of m and the localness of end a m. In this paper characterization of pseudo mp projective modules and quasi pseudo principally projective modules are given and discussed the various properties of it. Perhaps the most successful invariant measuring their complexity is the invariant. Generalization of semiprojective modules semantic scholar. 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. Then m is projective if and only if m b is a projective as a bmodule, of course. For example, a free resolution of a module m is a left resolution in which all the modules e i are free rmodules. A ring is called left hereditary if all left ideals.
A sufficient condition for a small pseudo projective module to be a s. This ideal is countably generated and projective but not a direct sum of finitely generated submodules. A short exact sequence of amodules is a sequence of the form 0. There are rings such that as modules, and so they do not have a unique dimensionrank.
M to indicate that n is an essential submodule, a small. 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. Recall that a submodule n of a module a is called small pseudo stable, if for any. Then there is a natural equivalence between addm and the category of projective right smodules. 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. Talebi and others published on pseudoprojective and pseudosmallprojective modules find, read and cite all the research you need on researchgate. Talebi and others published on pseudo projective and pseudosmallprojective modules find, read and cite all the research you need on researchgate. Bbe faithfully at, and ma nite amodule anoetherian. Various equivalent characterizations of these modules appear below. It is proved that a pseudo mpprojective module is hop.
An module is projective if and only if there exists an module such that is free proof. It is also provided the sufficient condition for pseudo mp. International journal of computer applications 0975 8887. 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. Likewise, projective and flat resolutions are left resolutions such that all the e i are projective and flat rmodules, respectively. Every free module is a projective module, but the converse fails to hold over some rings, such as dedekind rings. Notice that z2z and z3z are z6zmodules and we have an isomorphism of z6zmodules. Recall that a ring r is called quasifrobeinus if it is left. In this paper characterization of pseudo mpprojective modules and quasi pseudo principally projective modules are given and discussed the various properties of it. The study of the relation between x and the topology of x and its. Wecallamodule m tobeadualautomorphisminvariant module if whenever k1 and k2 are small submodules.
Note that mis projective i for all exact sequences 0. Projective modules and embedded noncommutative spaces 5 given an integer m n, we let lam. 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. Throughout is a ring with 1 and all modules are left modules corollary 1. Some results on pseudo semi projective modules in this section, we study some properties of pseudo semi projective module and its endomorphism ring. Clearly, every quasiprojective module is pseudoprojective. On pseudoprojective and pseudosmallprojective modules hikari.
Projective modules over dedekind domains, february 14, 2010. Pseudo principally quasiinjective modules and pseudo. An module is projective if and only if there exists an module such that is free. Projective modules with finitely many generators are studied in algebraic theory. In many circumstances conditions are imposed on the modules e i resolving the given module m. In other words, projective modules are the way to express vector bundles in algebraic language. 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. The following fact shows how to convert addm into a category of projective modules. In this paper, we introduce the concept of esmall mprojective modules as a. On ding injective, ding projective and ding flat modules and complexes gillespie, james, rocky mountain journal of mathematics, 2017. Then m is called projective if for all surjections p.
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. 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. 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. Likewise, projective and flat resolutions are left resolutions such that all the e i are projective and flat r modules, respectively. Semiprojective module, pseudo principally projective. 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.
Then m is finitely generated iff trm is finitely generated. For example, all free modules that we know of, are projective modules. In particular p is pseudo nite provided it is projective in. Pdf on semiprojective modules and their endomorphism rings. 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. Introduction the aim of this note is to construct left rmodules m, where ris a kalgebra over some eld k, with the following properties. Now we localize at a maximal ideal m and since this is. First of all, the structure of quasiprincipally projective modules was. The modules which are constructed will have finite length three and only three nonzero proper submodules.
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. In addition, a splitting property for projective modules recently established by gabber, liu and lorenzini is also discussed. 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. Symplectic topology of projective manifolds with small. These modules had to do with splitting of certain exact sequences. On image summand coinvariant modules and kernel summand. The simplest example of a projective module is a free module. The coincidence of the class of projective modules and that of free modules has been proved for. On pseudo semiprojective modules academic journals. The \only if follows from the same argument as the easy direction of.
We do not expect that every pseudo nite module is pself. In previous lecture notes, one can nd the theory of injective and projective modules. Projective modules are recognized by having trivial invariant, 2, 3. This paper introduces the notion of dual ofsuchmodules.
Semisimple modules fix a ring rnot necessarily commutative. Projective modules are important for at least the following reasons. In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules that is, modules with basis vectors over a ring, by keeping some of the main properties of free modules. See the history of this page for a list of all contributions to it. Iwasawa theory, projective modules, and modular representations. Let m be an r module, a submodule k of m is said to be. Section 3, we investigate radical projective modules. We have provide an example of module which are semiprojective but not quasiprojective. In this paper we generalize the basic properties of pseudo weakly projective modules. An algebraic formulation is given for the embedded noncommutative spaces over the moyal algebra developed in a geometric framework in 8. 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. Example of a projective module which is not a direct sum. Pseudo projective module, quasi projective module, quiver and representation 1. In this paper we construct pseudo projective modules which are not quasi projective over noncommutative perfect rings.
It is proved that a pseudo mp projective module is hop. 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. However, in the case of a finite dimensional algebra like this matrix ring, its easy to argue via dimensions that the rank is unique. In this paper, we have studied the properties of semiprojective module and its endomorphism rings related with hopfian, cohopfian, and directly finite modules. We explicitly construct the projective modules corresponding to the tangent bundles of the. The solution appears to rely on the reasoning of, but it isnt clear why one could reason with dimensions this way. In this thesis, we study the theory of projective and injective modules. Endomorphism rings of small pseudo projective modules. Projective and injective modules hw pushouts and pull. If ris a pid then every nitely generated projective rmodule is free. A module mr is called pseudo fqinjective or pfqinjective for short if every monomorphism from a finitely generated submodule of m to m. Over rings decomposable into a direct sum there always exist projective modules different from free ones. Pseudo principally quasiinjective modules and pseudo quasi. 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.
It is proved that a pseudo mpprojective module is hopfian iff mn is hopfian, for each fully invariant small submodule n of m. In analogy with the terminology local in the commutative case, the algebra a is called local if a. By remark 2 in part 1, there exists an exact sequence where is free. Z2z z3z thus z2z and z3z are nonfree modules isomorphic to direct summands of the free. Irving kaplansky, projective modules, the annals of mathematics second series, vol. Pseudo projective modules which are not quasi projective and quivers deste, gabriella and tutuncu, derya keskin, taiwanese journal of mathematics, 2018 euclid. If is free, then is projective by lemma 2 in part 1. Cinjectivity and cprojectivity yue chi ming, roger, hiroshima mathematical journal, 2007.
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. 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. For example, in most treatments i know the the definition of projective module is given as either. Throughout this paper all rings are associative rings with identity, and all. It is a long held belief in physics that the notion of spacetime as a pseudo riemannian manifold requires modi. Projective and injective modules hw pushouts and pullbacks.
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