An module is projective if and only if there exists an module such that is free. In this paper, we introduce the concept of esmall mprojective modules as a. 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. The following fact shows how to convert addm into a category of projective modules. So far we have only given a trivial example of projective modules, i. International journal of computer applications 0975 8887. On pseudoprojective and pseudosmallprojective modules hikari. Every free module is a projective module, but the converse fails to hold over some rings, such as dedekind rings.
These modules are termed as simpledirectinjective and simpledirectprojective, respectively. 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. 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. Endomorphism rings of small pseudo projective modules. Any pseudoprojective module is dual automorphisminvariant. Projective modules with finitely many generators are studied in algebraic theory. The invariant factors of a flat module being at each prime either 1 or 0 it implies for one such, say, that p is flat.
Indecomposability of m and the localness of end a m. On pseudo semiprojective modules academic journals. Pseudo projective and essentially pseudo injective modules 419 proposition 7. Pdf this paper provides the several homological characterization of perfect rings and semisimple rings in terms of semiprojective modules. A perfect complex is a pseudocoherent complex of finite tor dimension. Pseudo projective modules which are not quasi projective.
Projective modules and embedded noncommutative spaces 5 given an integer m n, we let lam. Notice that z2z and z3z are z6zmodules and we have an isomorphism of z6zmodules. An algebraic formulation is given for the embedded noncommutative spaces over the moyal algebra developed in a geometric framework in 8. Pseudo injective and essential pseudo injective modules have been studied by many authors 2, 3, but here we study the properties of pseudo and smallpseudo 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. Note that pseudo projective modules are named as epiprojective in 2. Pseudo injectivity is a generalization of quasi injectivity. 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.
N p be the projection map, m ms be the natural map. The coincidence of the class of projective modules and that of free modules has been proved for. Given the myriad projective modules which can be constructed by the method discussed in the first three sections, it is essential to have a way of classifying these modules. By m pseudo projectivity of n there exists a homomorphism h. In this paper, the concept of quasipseudo principally injective modules is introduced and a characterization of commutative semisimple rings is given in terms of quasipseudo principally injectiv. Pdf on semiprojective modules and their endomorphism rings. We do not expect that every pseudo nite module is pself. Introduction the aim of this note is to construct left rmodules m, where ris a kalgebra over some eld k, with the following properties. The \only if follows from the same argument as the easy direction of. Pseudo projective module, pseudo weakly projective module, projective cover. M if it is improjective with respect to all epis in. Recall that a ring r is called quasifrobeinus if it is left.
The annihilator of in is defined by we say that is faithful if. Bbe faithfully at, and ma nite amodule anoetherian. Now we localize at a maximal ideal m and since this is. Likewise, projective and flat resolutions are left resolutions such that all the e i are projective and flat r modules, respectively. Because z is a pid, q is also a free zmodule but its not. 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. An rmodule mis torsionfree if for all nonzerodivisors r2rand elements x2m, rx 0 if and only if x 0. Since finite projective modules can be characterized as summands of finite free modules algebra, lemma 10. On semiprojective modules and their endomorphism rin gs p t be a polynomial ring generated b y the transformati on t over the.
Clearly, every quasiprojective module is pseudoprojective. If e e 2 is an idempotent in the ring r, then re is a projective left module over r. Throughout this paper all rings are associative rings with identity, and all. Instead of pseudoprojective in rmod we will just say pseudoprojective for short. Similarly, the group of all rational numbers and any vector space over any eld are examples of injective modules. 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. Let m be an rmodule, a submodule k of m is said to be. The following is a well known lemma due to johnson and wong. Then there is a natural equivalence between addm and the category of projective right smodules. 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.
Direct sums and direct summands of projective modules are projective. 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. When the category is abelian, it is called a tensor category, following de1. Note that pseudo projective modules are named as epi projective in 2. Then m is finitely generated iff trm is finitely generated. M is said to be quasi projective pseudo projective if, for any submodule xof m, any homomorphism epimorphism f.
So by contradiction, suppose q is a projective zmodule. Recall that a submodule n of a module a is called small pseudo stable, if for any. Pseudo projective module, quasi projective module, quiver and representation 1. Talebi and others published on pseudo projective and pseudo small projective modules find, read and cite all the research you need on researchgate. 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. If q is a submodule of some other left rmodule m, then there exists another submodule k of m such that m is the internal direct sum of q and k, i. A left module q over the ring r is injective if it satisfies one and therefore all of the following equivalent conditions. If is free, then is projective by lemma 2 in part 1. Pseudo code practice problems queen annes county public. It is proved that a pseudo mpprojective module is hop. Oct 28, 2011 throughout is a ring with 1 and all modules are left modules. Talebi and others published on pseudoprojective and pseudosmallprojective modules find, read and cite all the research you need on researchgate.
Then m is projective if and only if m b is a projective as a bmodule, of course. On pseudoinjective and pseudoprojective modules 59 following 17 we say that a submodule n. A short exact sequence of amodules is a sequence of the form 0. Some results on pseudo semiprojective modules in this section, we study some properties of pseudo semiprojective module and its endomorphism ring. A klinear symmetric rigid monoidal karoubian category in which the endomorphisms of the tensor identity 1 constitute kwill be called a pseudotensor category. It is a long held belief in physics that the notion of spacetime as a pseudo riemannian manifold requires modi. For example, a free resolution of a module m is a left resolution in which all the modules e i are free r modules.
An english translation of this book is coming soon. In this paper characterization of pseudo mpprojective modules and quasi pseudo principally projective modules are given and discussed the various properties of it. Pdf in this paper characterization of pseudo mpprojective modules and quasi pseudo principally projective modules are given and discussed the. Projective modules over dedekind domains, february 14, 2010 5 iii ii. On pseudoprojective and essentially pseudo injective modules. For example, all free modules that we know of, are projective modules. The crucial tool which we use for this classification is the generalized chern character introduced by connes 8. Tania gabriela perezquijano, ivan fernando vilchis. Summer school and conference mathematics, algorithms. Robert wisbauer on module classes closed under extensions. A homomorphism of a pseudo plane onto a projective plane. Let m be a small pseudo projective module then m is s. By remark 2 in part 1, there exists an exact sequence where is free. A perfect module is a module that is perfect when it is viewed as a complex concentrated at degree zero.
A right rmodule m is called semiprojective if, for any submodule n of m, every epimorphism. Pseudoprojective modules u in rmod can be characterized by their trace ideals. In analogy with the terminology local in the commutative case, the algebra a is called local if a. In this paper we generalize the basic properties of pseudo weakly projective modules. Projective modules over dedekind domains, february. Over rings decomposable into a direct sum there always exist projective modules different from free ones. In many circumstances conditions are imposed on the modules e i resolving the given module m. Lectures on injective modules and quotient rings,lecture notes in mathematics 49. A module m is quasiinjective if and only if it is fully invariant in its injective hull em. Some results on pseudo semi projective modules in this section, we study some properties of pseudo semi projective module and its endomorphism ring. Note that mis projective i for all exact sequences 0.
In this paper we construct pseudo projective modules which are not quasi projective over noncommutative perfect rings. The simplest example of a projective module is a free module. Courter, finite direct sums of complete matrix rings over perfect completely primary rings,canad. We explicitly construct the projective modules corresponding to the tangent bundles of the. Two rmodules u and m are called trace equivalent if genu genm. The following properties of projective modules are quickly deduced from any of the above equivalent definitions of projective modules. Z2z z3z thus z2z and z3z are nonfree modules isomorphic to direct summands of the free. Pseudo projective modules which are not quasi projective and.
On pseudoprojective and pseudosmallprojective modules. Listed below is a brief explanation of pseudo code as well as a list of examples and solutions. We will not use this as the definition, but define perfect complexes over a ring directly as follows. Pseudo projective modules which are not quasi projective and quivers deste, gabriella and tutuncu, derya keskin, taiwanese journal of mathematics, 2018 pseudo diagrams of knots, links and spatial graphs hanaki, ryo, osaka journal of mathematics, 2010. Corational extensions and pseudoprojective modules. For example, if a is noetherian, a module over a is perfect if and only if it has finite. 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. Since any projective module isomorphic to a direct summand of a free module, it su ces to.
Dually, we give characterizations of those rings such that every right rmodule is nd projective. T h e n p t is a commutative noetherian r ing with. Various equivalent characterizations of these modules appear below. In this paper, we give a complete characterization of the aforementioned modules over the ring. In this thesis, we study the theory of projective and injective modules. In addition, a splitting property for projective modules recently established by gabber, liu and lorenzini is also discussed. A sufficient condition for a small pseudo projective module to be a s. Projective and injective modules arise quite abundantly in nature.
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