For a module m, the class of those modules n for which m is almost ninjective, is not closed under direct sums. 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. In particular, if n is indecomposable, then the diagram 2 gives that for every f. We provide a negative answer to a question raised by jain and alahmadi asking for a possible baerlike criterion for almost injective modules. Indecomposable decompositions of finitely presented pureinjective modules.
The author has defined almost projective modules and almost injective modules in 8, and by making use of the concept. It is proved that r is a right almost v ring if and only if for every rmodule m, any complement of every. 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 ninjective, where n is some module, was introduced by baba 1989.
A short exact sequence of amodules is a sequence of the form 0. A note on almost injective modules 103 proposition 3. 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. Projective and injective modules arise quite abundantly in nature. Baba gave a necessary and sufficient condition under which a uniform.
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. Septimiu crivei injective modules relative to torsion. Let rbe a ring with identity and let jbe an rmodule. We describe rings over which every right module is almost injective. For a given module m, the class of modules n, for which m is almost ninjective, is not closed under direct sums. We introduce the concepts of almostinjective and radinjective modules. A left rmodule m is called almost finjective, if every rhomomorphism from a finitely presented left ideal to m extends to a homomorphism of.
The regular and ifinite rings over which every right module is almost projective are described. 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. Almost relative projective modules and almost relative injective modules, monograph. Radinjective and almostinjective modules and rings. Enough injectives university of california, berkeley. 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.
A module m that is almost m injective, is called an almost self injective module. 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. Sam evens these notes are supplementary to class discussion, and much of them is in ash, 10. Pdf a note on almost injective modules researchgate. A module is called coneat injective if it is injective with respect to all coneat exact sequences. Equivalently, for every semisimple submodule k of n. Uniform almost relative injective modules sciencedirect. A module m that is almost minjective is called an almost selfinjective module.
Then the category of rmodules has enough injectives. 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. On almost projective and almost injective modules let a be an artin algebra, and consider the category moda of. Injective dimension measures how far from injective a module is. An injective module over r r is an injective object in r mod r mod. Pdf almost relative injective modules researchgate. The concept of a module m being almost ninjective where n is some module, was introduced by baba 1989. Every indecomposable injective module has a local endomorphism ring. Osaka university and osaka city university, departments of mathematics editorial board.
Let r r be a commutative ring and c r mod c r mod the category of r rmodules. 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. 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. 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. Theorem 5 shows that the endomorphism ring of an indecomposable almost self injective module is local. Almost finjective modules and almost flat modules 119 4 by proposition 2. Theorem 3 for any ring r, the category of rmodules has enough injectives. Similarly, the group of all rational numbers and any vector space over any eld are examples of injective modules. Relative cotilting theory and almost complete cotilting. 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. Almost finjective modules and almost flat modules 117 6 m is an almost pure submodule of its injective envelope em. The category a has left almost split sequences if for every indecomposable not injective a.
In the language of faithful dimension this means that no injective module m, with m pdf. For example, all free modules that we know of, are projective modules. When ndomdima 1, there is a minimal faithful strongly projectiveinjective module ae whose endomorphism ring eae is called the associated selfinjective subalgebra of a. Almost injective modules lack a baerlike criterion. Derived equivalences, restriction to selfinjective. A module m in moda is called almost projective if ext1 a m,x 6 0 for at most. The class of such modules is enveloping and falls properly between injectives and pure injectives. We prove that a ring r is semisimple if and only if the rmodule rr is falmost injective.
We determine the pureinjective hull of every direct sum string module over a string algebra. Then ris right almost selfinjective if and only if for every c. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. Suppose there is a be an injective rmodule j with the property that h jm 6 0 whenever m6 0. In this note, we study precovers and preenvelopes by f pinjective and f pflat modules, including their properties under almost excellent extensions of rings. Let rbe a commutative noetherian ring with an identity element. Pdf a note on almost injective modules adel alahmadi.
M is called almost selfinjective if m is almost minjective. We will rst establish the following preliminary result. The structure of rings over which every right module is almost injective is clarified. Projective and injective modules play a crucial role in the study of the cohomology of representations. Harada, on almost relative injectives on artinian modules, osaka j. For r r a ring, let r r mod be the category of r rmodules. 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. Noetherian rings with almost injective simple modules. On almost relative injectives on artinian modules manabu harada received. An rmodule jis an injective module if jsatis es one of the equivalent conditions of proposition46. A module m that is almost minjective, is called an almost selfinjective module.
Using these concepts, we provide some new characterizations of vrings, semilocal rings and pseudofrobenius rings. Title almost relative injective modules authors singh. 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. For a module m, the class of those modules n for which m is almost n injective, is not closed under direct sums. An almost property of a map gets sent to the actual property by j. A ring r is called right almost selfinjective if it is almost selfinjective as a right module over itself. A ring r is called a right almost v ring if every simple rmodule is almost injective. A ring r is called right almost selfinjective if it is almost self injective as a right module over itself. Remarks on f p injective and f p flat modules springerlink. M is almost injective if it is almost injective relative to all modules n. It is shown that each almost maximal valuation ring r, such that every indecomposable injective rmodule is countably generated, satis. For a given module m, the class of modules n, for which m is almost ninjective, is not.
A rmodule eis called injective if for each injective homomorphism f. Ssinjective modules and rings iraq academic scientific journals. 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. It is proved that r is such a ring, if and only if, the complements of. We characterize right noetherian rings over which all simple modules are almost injective.
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