A rmodule eis called injective if for each injective homomorphism f. The structure of rings over which every right module is almost injective is clarified. Remarks on f p injective and f p flat modules springerlink. Suppose there is a be an injective rmodule j with the property that h jm 6 0 whenever m6 0. Generalizations of coneat injectivity, like relative coneat injectivity and full invariance of a module in its coneat injective envelope, are studied. The concept of a module m being almost n injective, where n is some module, was introduced by baba 1989. The class of such modules is enveloping and falls properly between injectives and pure injectives. The articles are devoted to leavitt path algebras, roots of elements in lie groups, hilberts nullstellensatz, mixed multiplicities of ideals, singular matrices, rings of integers, injective hulls of modules, representations of linear, symmetric groups and lie algebras, the algebra of generic matrices and almost injective modules. Septimiu crivei injective modules relative to torsion. Let rbe a commutative noetherian ring with an identity element. The regular and ifinite rings over which every right module is almost projective are described. It is shown that each almost maximal valuation ring r, such that every indecomposable injective rmodule is countably generated, satis. Preparatory material for the snowbird summer school on commutative algebra these notes are intended to give the reader an idea what injective modules are, where they show up, and, to a small extent, what one can do with them.
Noetherian rings with almost injective simple modules. M is almost injective if it is almost injective relative to all modules n. Almost finjective modules and almost flat modules 117 6 m is an almost pure submodule of its injective envelope em. Every indecomposable injective module has a local endomorphism ring. Let r r be a commutative ring and c r mod c r mod the category of r rmodules. A module m in moda is called almost projective if ext1 a m,x 6 0 for at most. Relative cotilting theory and almost complete cotilting. For example, all free modules that we know of, are projective modules. An exact sequence 0 a f b g c 0ina is an almost split sequence if f is a left almost split morphism and g is a right almost split morphism. Then the category of rmodules has enough injectives.
Pdf a note on almost injective modules adel alahmadi. An injective module over r r is an injective object in r mod r mod. Moreover, the endo morphism ring of a uniserial almost. On almost projective and almost injective modules let a be an artin algebra, and consider the category moda of.
Theorem 5 shows that the endomorphism ring of an indecomposable almost self injective module is local. Theorem 3 for any ring r, the category of rmodules has enough injectives. An almost property of a map gets sent to the actual property by j. We determine the pureinjective hull of every direct sum string module over a string algebra. Baba gave a necessary and sufficient condition under which a uniform finite length module u is almost vinjective, where v is a finite direct sum of uniform, finite length. We introduce the concepts of almostinjective and radinjective modules. For a module m, the class of those modules n for which m is almost n injective, is not closed under direct sums. 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.
Derived equivalences, restriction to selfinjective. A module is called coneat injective if it is injective with respect to all coneat exact sequences. The category a has left almost split sequences if for every indecomposable not injective a. A module m that is almost minjective is called an almost selfinjective module. Injective dimension measures how far from injective a module is. If m and n are right rmodules, m is called socleninjective socninjective if every rhomomorphism from the socle of n into m extends to n. Osaka university and osaka city university, departments of mathematics editorial board.
It is proved that r is such a ring, if and only if, the complements of. Using these concepts, we provide some new characterizations of vrings, semilocal rings and pseudofrobenius rings. Almost projective and almost injective modules springerlink. Almost relative projective modules and almost relative injective modules, monograph. In particular, if n is indecomposable, then the diagram 2 gives that for every f. Ssinjective modules and rings iraq academic scientific journals. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. A note on almost injective modules 103 proposition 3. It is proved that r is a right almost v ring if and only if for every rmodule m, any complement of every. Pdf a note on almost injective modules researchgate. Pdf almost relative injective modules researchgate. Theorem 5 shows that the endomorphism ring of an indecomposable almost selfinjective module is local. For a given module m, the class of modules n, for which m is almost ninjective, is not. We give some new properties of almost injective modules and their endomorphism rings, and also provide conditions as to when a direct sum of almost injective or cs modules is again almost injective or cs in some special cases, n1 is a nonzero.
I think you will have more luck learning nontrivial facts about and examples of injective modules in noncommutative algebra texts rather than homological algebra texts. The concept of a module m being almost ninjective where n is some module, was introduced by baba 1989. A ring r is called a right almost v ring if every simple rmodule is almost injective. For r r a ring, let r r mod be the category of r rmodules. Almost qf rings and almost qr rings manabu harada received june 2, 1992 in this paper we assume that every ring r is an associative ring with identity and r is twosided artinian.
Enough injectives university of california, berkeley. For a given module m, the class of modules n, for which m is almost ninjective, is not closed under direct sums. Pdf we give some new properties of almost injective modules and their endomorphism rings, and also provide conditions as to when a direct sum of. Almost injective modules lack a baerlike criterion. Almost finjective modules and almost flat modules 119 4 by proposition 2.
An rmodule jis an injective module if jsatis es one of the equivalent conditions of proposition46. When ndomdima 1, there is a minimal faithful strongly projectiveinjective module ae whose endomorphism ring eae is called the associated selfinjective subalgebra of a. It is shown that there exists an rmodule m which is almost injective relative to r but not almost injective. Let rbe a ring with identity and let jbe an rmodule. For a given module m, the class of modules n, for which m is almost nm, the class of modules n, for which m is almost n. Title almost relative injective modules authors singh. Baba gave a necessary and sufficient condition under which a uniform. We give some new properties of almost injective modules and their endomorphism rings, and also provide conditions as to when a direct sum of almost injective or cs modules is again almost. B with b and ba injective, then a is nearly injective itself it has injective dimension 1, and so many arguments for a merely have to combine arguments for b and ba, both of which are supposed to. We describe rings over which every right module is almost injective.
In homological algebra texts, injective modules and more generally injective objects in an abelian category play an important technical role but are not really studied for their own sake. Uniform almost relative injective modules sciencedirect. We characterize right noetherian rings over which all simple modules are almost injective. Sam evens these notes are supplementary to class discussion, and much of them is in ash, 10. Radinjective and almostinjective modules and rings. For a module m, the class of those modules n for which m is almost ninjective, is not closed under direct sums. In this note, we study precovers and preenvelopes by f pinjective and f pflat modules, including their properties under almost excellent extensions of rings. We will rst establish the following preliminary result. Equivalently, for every semisimple submodule k of n. A module m that is almost m injective, is called an almost self injective module. A short exact sequence of amodules is a sequence of the form 0. In addition, we also introduce and investigate f pprojective modules.
The author has defined almost projective modules and almost injective modules in 8, and by making use of the concept. A left rmodule m is called almost finjective, if every rhomomorphism from a finitely presented left ideal to m extends to a homomorphism of. We provide a negative answer to a question raised by jain and alahmadi asking for a possible baerlike criterion for almost injective modules. A module m that is almost minjective, is called an almost selfinjective module.
The concept of a module m being almost ninjective, where n is some module, was introduced by baba 1989. M is called almost selfinjective if m is almost minjective. Then ris right almost selfinjective if and only if for every c. Harada, on almost relative injectives on artinian modules, osaka j. Projective and injective modules play a crucial role in the study of the cohomology of representations. Then there exists an almostinjective function from x to y in particular, the constant function onto y 1 but even though y is t 2, x is not. Left almost selfinjective rings are defined, similarly.
Projective and injective modules arise quite abundantly in nature. Indecomposable decompositions of finitely presented pureinjective modules. In the language of faithful dimension this means that no injective module m, with m pdf. On almost relative injectives on artinian modules manabu harada received.