Example of a projective module which is not a direct sum. Projective modules over universal enveloping algebras september 27, 2011 3 14. So, we recall some fundamental results about the gorenstein projective modules and dimensions. Let m i be a collection of left r modules indexed by the set i, and let n be a left r module. Exactness edit an r module p is projective if and only if the covariant functor hom p. Section 1 introduces basic facts about projective modules and some concrete examples. This paper extends and simplifies this result further and shows that an injective module m is. By a theorem of kaplansky every projective module is a direct sum of countably generated ones. We have discussed the summand intersection property, summand sum property for semi projective modules. For example, all free modules that we know of, are projective modules. Introduction one of the fundamental tools to describe the direct sum decompositions of a module is to study the projective modules over its endomorphism ring.
First, a direct sum of r modules is projective iff each one is projective. It is a wellknown and basic result of homological algebra that the direct product of an arbitrary family of injective modules over any ring is again injective 3, p. An affirmative answer would resolve several major open problems, as explained in p. On semiprojective modules and their endomorphism rings. But i need an example to show that, an arbitrary direct product of projective modules need not be a projective module. An example of a pseudo mprincipally projective module which is not m projective is given. Stack exchange network consists of 175 q&a communities including stack overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers visit stack exchange. The direct sum of every pair of quasi projective modules is quasi projective. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. If sis nite, then there is no di erence between the direct sum and the direct product. As a third application we record the following result, proved in 5.
Projective and injective modules play a crucial role in the study of the cohomology of representations. The coincidence of the class of projective modules and that of free modules has been proved for. We study direct sum decompositions of rd projective modules. Some of the methods are modeltheoretic, and the techniques developed using these may be of interest in their own right. For a projective amodule p, the following are equiva lent. One checks that this gives an isomorphism of z 6modules z6. We characterize rings over which every projective module is a direct sum of. Request pdf direct sums of injective and projective modules it is well known that a countably injective module is. Rings which have a projective cover for each module are called left perfect rings, and such rings have been characterized by h. In this case, we have additional characterizations of projective modules.
Pardos paper separative cancellation for projective modules over exchange rings, israel j. Some consequences and generalizations are also obtained. 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. Over rings decomposable into a direct sum there always exist projective modules different from free ones. While projective covers for modules do not always exist, it was speculated that for general rings, every module would have a flat cover, that is, every module m would be the epimorphic image of a flat module f such that every map from a flat module onto m factors through f, and any endomorphism of f over m is an automoprhism. The simplest example of a projective module is a free module. As a corollary we deduce that every projective module over a commutative bezout ring of. In addition, we will see the connection between the divisor class group and the picard group. Projective modules over universal enveloping algebras september 27, 2011 2 14. Their direct sum i2im i is the set of all tuples x i i2i such that x i 2m i for all i2i and all but nitely many x. Rings over which every rdprojective module is a direct. So z 2 and z 3 are projective but not free z 6 modules. Let r z6 and let m be the principal ideal of rgenerated by 3, so m. We show that every free projective right r module is baer if and only if ris a right semiprimary, right hereditary.
The direct sum s2sm sis the submodule of q s2s m sgiven by the condition that all but nitely many coordinates are zero. R mod ab is an exact functor, where r mod is the category of left r modules and ab is the category of abelian groups. International journal of computer applications 0975 8887. Isomorphism between direct sum of modules mathoverflow. Hence, every module over a semisimple ring is projective. A module p is projective if and only if there is another module q such that the direct sum of p and q is a free module. For a finite projective module over a commutative local ring, the theorem is an easy consequence of nakayamas lemma. The submodule of i m consisting of all elements m such that m 0 for all but finitely many components m is called the direct sum of the modules m i, and is denoted by i m. Direct sums of vector spaces thursday 3 november 2005 lectures for part a of oxford fhs in mathematics and joint schools direct sums of vector spaces projection operators idempotent transformations two theorems direct sums and partitions of the identity important note.
Direct sums and direct products of finitedimensional. Every projective module is a direct sum of countably generated projective submodules. It is proved that this is the case over hereditary noetherian rings. In general, modules with this property are called projective. Projective modules over universal enveloping algebras september 27, 2011. Direct sums of injective and projective modules request pdf.
Section 2 presents some classical structure theorems for projective modules established around the 60s and 70s. Apart from this some results are proved related to hop. For a direct sum this is clear, as one can inject from or project to the summands. While the structure of pure projective modules is more attractive than. A nonzero r module m is called simple if it has no submodules other than 0 and m, and semisimple if it is the direct sum of simple modules. Observe that a projective module over an arbitrary ring is a direct sum of countably generated projective modules. A module is pure projective precisely if it is a direct sum of finitedimensional submodules, while direct summands of direct products of finitedimensional modules are the pureinjective modules, see 16 and 14, respectively, or 19. Direct sums of injective and projective modules core. It is proved that if r is a right ore domain with right. Then for every there exists an module and a free module such that let and then is a free module and thus is projective. The question is addressed of when all pure projective modules are direct sums of finitely presented modules. In an earlier paper, we described the structure of rings over which every countably generated right module satisfies, and it was shown that such a ring is right artinian. For a xed element s 0 2sthe canonical injective map m s 0.
The first summand is free, so its projective resolution is rather simple. Let the module m m 1 m 2 be a direct sum of relatively projective modules m 1. Direct sums of quasiinjective modules are also investigated. It follows that every a module is a direct summand of a free a module. In particular, we investigate the rings over which every rd projective right module is a direct sum of cyclically presented right modules, or a direct sum of finitely presented cyclic right modules, or a direct sum of right modules with local endomorphism rings ssp rings. By the preivous item, the restriction to a sylow p subgroup h of a projective module is projective. A direct sum of r modules l i2i p i is projective i each p i is projective. Apart from this, we have introduced the idea of mhollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi projective modules. If rhas a dualizing complex, then any direct product of gorenstein. By the preivous item, the restriction to a sylow psubgroup h of a projective module is projective. Take the direct sum of projective resolutions of each summand.
Projective and injective modules arise quite abundantly in nature. A module m is said to satisfy the condition if m is a direct sum of a projective module and a quasicontinuous module. Projective modules with finitely many generators are studied in algebraic theory. Kaplanskys theorem on projective modules wikipedia. Then it is wellknown thatzand zp1 are relatively projective,mdoes not have d1 and zp1hasd1. Some results on direct sum decompositions of baer modules are also included. So you get a finite free resolution of length 1 which is normal since a p. The study of stably free modules has a rich history, and we cannot do it justice here. Request pdf direct sums of semi projective modules we investigate when the direct sum of semi projective modules is semiprojective.
An excellent source for further information is the book lam. The main results of this paper are in section 3, where we investigate the validity of the baer property for a direct sum of copies of a baer r module. Projective modules over universal enveloping algebras. For the general case, the proof both the original as well as later one consists of the following two steps. So the restriction is a direct sum of indecomposable projectives. Consider the z module m z zp1wherezp1 denotes the prufer pgroup.
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