For a xed element s 0 2sthe canonical injective map m s 0. Injective modules over noetherian rings mathematical sciences. Coneat injective modules hamid, mohanad farhan, missouri journal of mathematical sciences, 2019. So if c is a nonzero tensor ideal localising subcategory of modkg. A ring ris noetherian if and only if every direct sum of injective r modules is injective. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. Projectiveinjective modules, serre functors and symmetric.
Lannes t functor on injective unstable modules and harish. Contrast with the direct product, which is the dual notion. The cartesian product again gives the direct product, but the direct sum object must be constructed in a much more involved way called a free product. The results obtained include identification of the free, projective, and injective modules. So any ndimensional representation of gis isomorphic to a representation on cn. Torsionfree dimension of modules and selfinjective dimension of rings huang, chonghui and huang, zhaoyong, osaka journal of mathematics, 2012. Direct sums and direct products of finitedimensional. Direct sums of injective and projective modules request pdf. February 12, 2007, may 24, 2007, july 27, 2007, and september 15, 2007 published electronically. Edit thanks to jeremy rickard for several corrective insights in the comments in the direction of positive conditions, im not sure whether the following conditions are reasonable for your purposes. V 2 has an obvious structure of a representation of a.
Let r be a hereditary torsion theory with respect to r see golan 3. If sis nite, then there is no di erence between the direct sum and the direct product. Projective and injective modules play a crucial role in the study of the cohomology of representations. The direct sum of these representations is the direct sum of and as. Beachy department of mathematical sciences northern illinois university dekalb, il 60115 for a left rmodule m, we identify certain submodules of mthat play a role analogous to that of prime ideals in the ring r. There are two ways to think of representation theory. Projectiveinjective modules, serre functors and symmetric algebras volodymyr mazorchuk and catharina stroppel abstract we describe serre functors for generalisations of the category o associated with a semisimple complex lie algebra. If c is a class of right rmodules, a module m will be called sigma c, in case m is isomorphic to a direct sum of modules in c. Every injective module is uniquely a direct sum of indecomposable injective modules, and the indecomposable injective modules are uniquely identified as the injective hulls of the quotients rp where p varies over the prime spectrum of the ring. An analogous coordinatewise definition holds for infinite direct sums. In section 3, we show that a right rmodule m is strongly ss injective if and only if every small submodule a of a right rmodule n, every rhomomorphism a. On injective modules and support varieties for the small quantum group.
Scott grothendieck 7 showed that finite dimensional vector bundles over the projective line split into direct sum osf line bundles see also andersen 1 w. Projective and injective modules hw pushouts and pull. Thus, we see that there are exactly three isomorphism classes of indecomposable representations of q,0. Thus, if r is qf, then r is right left noetherian, so each direct sum of injective right r modules is injective. R is artinian you can prove this and any module m is a direct sum of indecomposable modules, injective if m is injective. While the structure of pureprojective modules is more attractive than that of pureinjective modules, both classes of modules be described by cardinal invariants, see 18. Definition in terms of linear representation as a module over the group ring. Injective modules for group algebras of locally finite groups. An injective module over a noetherian not necessarily commutative ring r is a possibly in nite direct sum of indecomposable injective modules. Then every rgtorsion free direct sum mand of a rgcompletely decomposable module is quasi injective see 4. Quotients of direct sums of modules physics forums. I would like to restrict my consideration to an arbitrary abelian subcategory c of the category rmod of unitary left r modules.
The direct sum of these representations is the direct sum of and as modules. It is proved that a module that is either reduced and locally finite or bounded below is a direct sum of cyclic submodules. On essential extensions of direct sums of injective modules. In our approach, projectiveinjective modules, that is modules which are both, projective and injective, play an important role. Browse other questions tagged abstractalgebra modules noetherian directsum injectivemodule or ask your own question. On injective modules and support varieties for the small. While the structure of pureprojective modules is more attractive than. Directsum representations of injective modules sciencedirect. Let v be an irreducible representation of a of dimension n. While the structure of pureprojective modules is more attractive than that of pure injective modules, both classes of modules be described by. A function is a way of matching the members of a set a to a set b. Coinduced representations and injective modules for hyperalgebra hr. On injective lmodules hindawi publishing corporation. We rst show that if mis a nitely generated rmodule, then hom r.
It is shown in 1, 3, 4 that r is a noetherian ring if and only if every injective module is. Pdf essential extensions of a direct sum of simple modules. It is shown that a semiregular ring r with the property that each essential extension of a direct sum of simple right rmodules is a direct sum of quasiinjective right rmodules is right noetherian. G, in which case is called a faithful representation. I of nonisomorphic simple right rmodules is a minimal right cogenerator for r. A ring ris noetherian if and only if every direct sum of injective rmodules is injective. In section 3, we show that a right rmodule m is strongly ssinjective if and only if every small submodule a of a right rmodule n, every rhomomorphism a. Pure submodules are direct summands if they are bounded below. That 5 implies 4 and that 2 implies 5, as well as the details of the others, may be looked up 2, p. Walker rutgers university, new brunsviick, new jersey, new mexico state university, university park, new mexico, and instituteor advanced study, princeton, new jersey received october 25, 1965 one of the main results of this article is the following. Representation theory university of california, berkeley. Hi, i keep seeing indirect uses of a result which i think would be stated as follows. Walker, directsum representations of injective modules. Pdf injective modules over wnoetherian rings, ii researchgate.
Baer, abelian groups that are direct summands of every containing abelian group, bulletin ams 46 no. 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. If is injective, then g g, in which case is called a faithful representation. Stratifying modular representations 1645 and an in nite dimensional version of dades lemma, which play no role in our.
Direct sum of injective modules is injective mathoverflow. A nonzero representation v of ais said to be irreducible if its only subrepresentations are 0 and v itself, and indecomposable if it cannot be written as a direct sum of two nonzero subrepresentations. Minimal cogenerators over osofsky and camillo rings. Direct sum representations of rings and modules springerlink. The direct sum of modules is the smallest module which contains the given modules as submodules with no unnecessary constraints, making it an example of a coproduct. In this case, r is also leftartinian by rosenbergzelinsky 20. Purity, algebraic compactness, direct sum decompositions, and. Injective, surjective and bijective injective, surjective and bijective tells us about how a function behaves. Representation theory of finite groups and associative. The tensor product of any module with kgis a direct sum of copies of kg. Direct sum of rinjective modules by kanzo masaike and takanori horigome throughout this paper r is a ring with identity and every z. Journal of algebra vol 5, issue 2, pages 3265 february. 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.
The direct sum s2sm sis the submodule of q s2s m sgiven by the condition that all but nitely many coordinates are zero. Zahedi and ameri 9 introduced the concepts of fuzzy projective and injective modules. Joubnal of algebra 5, 203221 1967 direct sum representations of injective modules carl faith and elbert a. Then c is contained in a direct sum of 8 the hypothesis implies that each injective module is a direct sum of finitely generated modules.
While the injective hull ec is the unique up to isomorphism minimal injective right cogenerator, osofsky 0 showed c is not necessarily unique even for commutative r, but that it is when r is either right noetherian, semilocal, or. Dec, 2004 torsionfree dimension of modules and self injective dimension of rings huang, chonghui and huang, zhaoyong, osaka journal of mathematics, 2012. Jan 03, 2015 the direct sum is the subset of the direct product consisting of those elements having only finitely many nonidentity coordinates. In particular, the so called generic module and the pru.
In theorem 3, we shall see that this result remains true, even if r is an arbitrary. Introduction to representation theory mit mathematics. Ams proceedings of the american mathematical society. The concept of free fuzzy modules was introduced by muganda 8, which is later generalized to that of free lmodules cf. Joubnal of algebra 5, 203221 1967 directsum representations of injective modules carl faith and elbert a. So z 2 and z 3 are projective but not free z 6 modules. The direct sum c of the injective hulls ev i of the set v i i. A semisimple or completely reducible representation of a is a direct sum of irreducible representations. Mackey, multiplicity free representations of finite groups. By the basspapp theorem, for a unital ring r, any direct sum of injective left r modules is injective if and only if r is left noetherian. Projective and injective modules hw pushouts and pullbacks. Direct sums and direct products of finitedimensional modules.
They control the serre functor in the case of a quasihereditary algebra having a double centraliser with respect to a projectiveinjective module whose endomorphism ring. A direct sum of r modules l i2i p i is projective i each p i is projective. In our approach, projectiveinjective modules, that is modules which are both, projective and injective. The use of an abstract vector space does not lead to new representation, but it does free us from the presence of a distinguished basis. Almost relative injective modules singh, surjeet, osaka journal of mathematics, 2016. In the 1980s, the magic properties of the cohomology of elementary abelian groups as modules over the steenrod algebra initiated a long lasting interaction between topology and modular representation theory in natural characteristic. Any direct summand of a module in annals of mathematics. Jan 21, 2012 hi, i keep seeing indirect uses of a result which i think would be stated as follows. On coverings of modules rim, seog hoon and teply, mark l. It is shown that a semiregular ring r with the property that each essential extension of a direct sum of simple right r modules is a direct sum of quasi injective right r modules is right noetherian. Thus to show that injectives are projective, it su. Coneat injective modules hamid, mohanad farhan, missouri journal of mathematical sciences, 2019 almost relative injective modules singh, surjeet, osaka journal of mathematics, 2016 gorenstein injective modules and ext. So if c is a nonzero tensor ideal localising subcategory of modkg, it contains. If r is a pid then a module m is injective if and only if it is divisible.
Cosemisimple modules and generalized injectivity liu, zhongkui and ahsan, javed, taiwanese journal of mathematics, 1999. The direct product of z and z for instance is the cartesian product zxz with the usual definition of coordinatewise operations, but the categorical direct sum is the free group on 2 generators. S eilenberg, t nakayamaon the dimension of modules and algebras. Representations of quivers university of california. A free module is a direct sum of copies of the left.
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