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. Hi, i keep seeing indirect uses of a result which i think would be stated as follows. Direct sum representations of rings and modules springerlink. Ams proceedings of the american mathematical society. It is shown in 1, 3, 4 that r is a noetherian ring if and only if every injective module is. Quotients of direct sums of modules physics forums. 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. Directsum representations of injective modules sciencedirect. Direct sum of rinjective modules by kanzo masaike and takanori horigome throughout this paper r is a ring with identity and every z. On injective lmodules hindawi publishing corporation. It is proved that a module that is either reduced and locally finite or bounded below is a direct sum of cyclic submodules.
Pdf injective modules over wnoetherian rings, ii researchgate. In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. Pure submodules are direct summands if they are bounded below. G, in which case is called a faithful representation. Definition in terms of linear representation as a module over the group ring. Coneat injective modules hamid, mohanad farhan, missouri journal of mathematical sciences, 2019. The direct sum s2sm sis the submodule of q s2s m sgiven by the condition that all but nitely many coordinates are zero. In this case, r is also leftartinian by rosenbergzelinsky 20. A function is a way of matching the members of a set a to a set b. Projective and injective modules play a crucial role in the study of the cohomology of representations. In our approach, projectiveinjective modules, that is modules which are both, projective and injective, play an important role.
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. Representations of quivers university of california. Pdf essential extensions of a direct sum of simple modules. There are two ways to think of representation theory. While the structure of pureprojective modules is more attractive than. 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. Furthermore, v0 0 is the direct sum of dimv0 copies of k0, v1 w0 is the direct sum of dimv1 copies of kkand.
Walker, directsum representations of injective modules. 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. 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. In particular, the so called generic module and the pru. 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. Projectiveinjective modules, serre functors and symmetric.
Direct sums and direct products of finitedimensional modules. Almost relative injective modules singh, surjeet, osaka journal of mathematics, 2016. 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. That 5 implies 4 and that 2 implies 5, as well as the details of the others, may be looked up 2, p. Joubnal of algebra 5, 203221 1967 direct sum representations of injective modules carl faith and elbert a. Thus, we see that there are exactly three isomorphism classes of indecomposable representations of q,0. Joubnal of algebra 5, 203221 1967 directsum representations of injective modules carl faith and elbert a.
We rst show that if mis a nitely generated rmodule, then hom r. Baer, abelian groups that are direct summands of every containing abelian group, bulletin ams 46 no. So any ndimensional representation of gis isomorphic to a representation on cn. So if c is a nonzero tensor ideal localising subcategory of modkg. Purity, algebraic compactness, direct sum decompositions, and. Direct sums and direct products of finitedimensional. 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 pull. 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 essential extensions of direct sums of injective modules. Thus, if r is qf, then r is right left noetherian, so each direct sum of injective right r modules is injective.
Thus to show that injectives are projective, it su. A direct sum of r modules l i2i p i is projective i each p i is projective. 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. Torsionfree dimension of modules and selfinjective dimension of rings huang, chonghui and huang, zhaoyong, osaka journal of mathematics, 2012.
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. Direct sum of injective modules is injective mathoverflow. R is artinian you can prove this and any module m is a direct sum of indecomposable modules, injective if m is injective. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. The direct sum c of the injective hulls ev i of the set v i i. If is injective, then g g, in which case is called a faithful representation. The direct sum of these representations is the direct sum of and as modules. Injective modules for group algebras of locally finite groups. Representation theory university of california, berkeley. 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. February 12, 2007, may 24, 2007, july 27, 2007, and september 15, 2007 published electronically.
So if c is a nonzero tensor ideal localising subcategory of modkg, it contains. Jan 21, 2012 hi, i keep seeing indirect uses of a result which i think would be stated as follows. A ring ris noetherian if and only if every direct sum of injective r modules 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. Any direct summand of a module in annals of mathematics. 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. 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.
Journal of algebra vol 5, issue 2, pages 3265 february. Zahedi and ameri 9 introduced the concepts of fuzzy projective and injective modules. While the structure of pureprojective modules is more attractive than that of pure injective modules, both classes of modules be described by. The results obtained include identification of the free, projective, and injective modules. Before proceeding further, we would like to introduce some terminology. Minimal cogenerators over osofsky and camillo rings. A semisimple or completely reducible representation of a is a direct sum of irreducible representations. Then every rgtorsion free direct sum mand of a rgcompletely decomposable module is quasi injective see 4. On injective modules and support varieties for the small quantum group. Jan 03, 2015 the direct sum is the subset of the direct product consisting of those elements having only finitely many nonidentity coordinates. 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.
On coverings of modules rim, seog hoon and teply, mark l. 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. 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. On injective modules and support varieties for the small. Cosemisimple modules and generalized injectivity liu, zhongkui and ahsan, javed, taiwanese journal of mathematics, 1999.
Projective and injective modules hw pushouts and pullbacks. Dec, 2004 torsionfree dimension of modules and self injective dimension of rings huang, chonghui and huang, zhaoyong, osaka journal of mathematics, 2012. If sis nite, then there is no di erence between the direct sum and the direct product. Stratifying modular representations 1645 and an in nite dimensional version of dades lemma, which play no role in our. Suppose has two linear representations over a field. In our approach, projectiveinjective modules, that is modules which are both, projective and injective. Lannes t functor on injective unstable modules and harish. Injective modules over noetherian rings mathematical sciences. V 2 has an obvious structure of a representation of a. S eilenberg, t nakayamaon the dimension of modules and algebras.
If r is a pid then a module m is injective if and only if it is divisible. 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. 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. The direct sum of these representations is the direct sum of and as. 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. A free module is a direct sum of copies of the left. 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. I would like to restrict my consideration to an arbitrary abelian subcategory c of the category rmod of unitary left r modules.
The concept of free fuzzy modules was introduced by muganda 8, which is later generalized to that of free lmodules cf. For a xed element s 0 2sthe canonical injective map m s 0. So z 2 and z 3 are projective but not free z 6 modules. Representation theory of finite groups and associative. In theorem 3, we shall see that this result remains true, even if r is an arbitrary. 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.
Mackey, multiplicity free representations of finite groups. 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. Contrast with the direct product, which is the dual notion. 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.
Let v be an irreducible representation of a of dimension n. Injective, surjective and bijective injective, surjective and bijective tells us about how a function behaves. Browse other questions tagged abstractalgebra modules noetherian directsum injectivemodule or ask your own question. An injective module over a noetherian not necessarily commutative ring r is a possibly in nite direct sum of indecomposable injective modules. Introduction to representation theory mit mathematics. Let r be a hereditary torsion theory with respect to r see golan 3. Direct sums of injective and projective modules request pdf. An analogous coordinatewise definition holds for infinite direct sums. Coinduced representations and injective modules for hyperalgebra hr.
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