Modules Invariant Under Clean Endomorphisms of Their Injective Hulls

We study modules which are characteristic submodules in their injective hulls. The author is supported by the Russian Foundation for Basic Research, project 08-01- 00693-a All rings below are assumed to be associative and with nonzero identity ele- ment; all modules are assumed to be unitary. A moduleM is said to be Noetherian if every properly ascending chain of submodules in M is finite. Expressions of the form 'A is a Noetherian ring' mean that AA and AA are Noetherian modules. A moduleM is said to be injective with respect to the moduleX or X-injective if for any submodule X1 in X, every homomorphismX1 → M may be extended to a homomorphismX → M. For a ring A, an A-module M is said to be injective if M is injective with respect to any A-module. A submodule M of the module E is said to be essential if M ∩ E1 6 0 for any nonzero submodule E1 in E. If E is an injective module and M is an essential submodule in E then E is called the injective hull of the module M. The injective hull is unique up to isomorphism. A submoduleM of the module E is called a characteristic submodule if f(M) ⊆ M for every automorphismf of the module E. In the papers (3), (8), and (10), a module M is said to be automorphism- invariant if M is a characteristic submodule of the injective hull of M. It is easy to verify that a module M is an automorphism-invariant if and only if M is a charac- teristic submodule of some injective module; see Remark 6 below. Remark 1. In (3, Theorem 16), it is proved that a module M is an automorphism-invariant module if and only if M is a pseudo-injective module, i.e., for any submodule X in M , every monomorphism X → M may be exten- ded to an endomorphism of the moduleM . For example, see (6), (11), (3) about pseudo-injective modules. Remark 2. It is clear that every injective module is an automorphism-invariant module. The converse is not true, since every simple Abelian group is a finite automorphism-invariant non-injective module over the ring of integers Z.

A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. In [Algebras for which every indecomposable right module is invariant in its injective envelope, Pacific J. Math., vol. 31, no. 3 (1969), 655-658] Dickson and Fuller had shown that if R R is a finite-dimensional algebra over a field F \mathbb F with more than two elements then an indecomposable automorphism-invariant right R R -module must be quasi-injective. In this paper we show that this result fails to hold if F \mathbb F is a field with two elements. Dickson and Fuller had further shown that if R R is a finite-dimensional algebra over a field F \mathbb F with more than two elements, then R R is of right invariant module type if and only if every indecomposable right R R -module is automorphism-invariant. We extend the result of Dickson and Fuller to any right artinian ring. A ring R R is said to be of right automorphism-invariant type (in short, RAI-type) if every finitely generated indecomposable right R R -module is automorphism-invariant. In this paper we completely characterize an indecomposable right artinian ring of RAI-type.

In this paper, we introduce the notion of essential extensions of fuzzy modules. We use these concepts to introduce the notion of injective hulls of fuzzy modules. It is known that every [Formula: see text]-module has an injective hull, where [Formula: see text] is a ring. We show that these corresponding results do not hold for fuzzy [Formula: see text]-modules, i.e. there exists a fuzzy [Formula: see text]-module that does not have an injective hull. Sufficient conditions are given for a fuzzy [Formula: see text]-module to have an injective hull.

The homological theory of rings and modules is an important branch of algebra. It provided answers to numerous questions of the theory of rings. Along with the homological theory, another theory started to develop, also under significant influence of the theory of rings, which is the homological theory of universal algebras, and, in particular, of semigroups and acts over them. This theory analyses such notions as injective and projective acts over semigroups, injective hulls and projective covers. As in the case of rings and modules, the injective hull exists for every act, while the projective cover sometimes does not. In 1967 P. Berthiaume proved the existence of injective hulls of an arbitrary act over a semigroup (without the assumption of the presence of an identity in the semigroup). J. Isbell studied monoids (i.e. semigroups with an identity) over which every act has a projective cover. L. A. Skornyakov developed a homological theory of monoids. Many results of that theory were mentioned in the known monograph by M. Kilp, U. Knauer, A. V. Mikhalev. For semigroups of a relatively simple structure the results of the homological theory can be significantly refined. For example, in 2012 G. Moghaddasi described injective acts and built injective hulls of acts over a left zero semigroup assuming the separability of the act. I. B. Kozhukhov and A. P. Haliullina described injective and projective acts over groups and right zero semigroups, built injective hulls and projective covers of acts over such semigroups. For acts over a left zero semigroup the condition of separability of acts was removed. An important class of semigroups containing groups, left and right zero semigroups, rectangular bands is the class of completely simple semigroups, as well as the broader class of completely 0-simple semigroups. In 2000 A. Yu. Avdeyev and I. B. Kozhukhov described all acts over completely simple semigroups and acts with zero over completely 0-simple semigroups. It triggered further reasearch of acts over such semigroups. I. B. Kozhuhov and A. O. Petrikov described injective and projective acts over completely simple semigroups, thereby generalising the results of I. B. Kozhuhov and A. R. Khaliullina, and also the work of G. Mogaddasi. They built injective hulls and projective covers of acts over such semigroups. In this paper the above-mentioned results concerning acts over completely simple semigroups were generalized to acts with zero over completely 0-simple semigroups. In particular, the necessary and sufficient conditions of injectivity and projectivity of an act with zero over an arbitrary completely 0-simple semigroup were found, injective hulls and projective covers of arbitrary acts with zero over such semigroups were built. It was established that a projective act over an arbitrary completely 0-simple semigroup is exactly a 0-coproduct of a free act and acts isomorphic to a 0-minimal right ideal of the semigroup (considered as a right act).

It is well known that the injective hull of a lattice-ordered group ( l l -group) M M can be given a lattice order in a unique way so that it becomes an l l -group extension of M M . This is not the case for an arbitrary f f -module over a partially ordered ring (po-ring). The fact that it is the case for any l l -group is used extensively to get deep theorems in the theory of l l -groups. For instance, it is used in the proof of the Hahn-embedding theorem and in the characterization of ℵ a {\aleph _a} -injective l l -groups. In this paper we give a necessary and sufficient condition on the injective hull of a torsion-free f f -module M M (over a directed essentially positive po-ring) for it to be made into an f f -module extension of M M (in a unique way). An f f -module is called an i − f i - f -module if its injective hull can be made into an f f -module extension. The class of torsion-free i − f i - f -modules is closed under the formation of products, sums, and Hahn products of strict f f -modules. Also, an l l -submodule and a torsion-free homomorphic image of a torsion-free i − f i - f -module are i − f i - f -modules. Let R R be an f f -ring with zero right singular ideal whose Boolean algebra of polars is atomic. We show that R R is a q f qf -ring (i.e., R R {R_R} is an i − f i - f -module) if and only if each torsion-free R − f R - f -module is an i − f i - f -module. There are no injectives in the category of torsion-free R − f R - f -modules, but there are ℵ a {\aleph _a} -injectives. These may be characterized as the f f -modules that are injective R R -modules and ℵ a {\aleph _a} -injective l l -groups. In addition, each torsion-free f f -module over R R can be embedded in a Hahn product of l l -simple Q ( R ) − f Q(R) - f -modules. We note, too, that a totally ordered domain has an i − f i - f -module if and only if it is a right Ore domain.

Let R M _RM be a left R R -module such that Hom R ( M , U ) ≠ 0 {\operatorname {Hom} _R}(M,U) \ne 0 for any nonzero submodule U U of M M , let E ( M ) E(M) denote the injective hull of M M , and let B B (resp. A A ) denote the ring of R R -endomorphisms of M M (resp. E ( M ) E(M) ). It is known that if M M is nonsingular then B B is left nonsingular and A A is the maximal left quotient ring of B B . We give here necessary and sufficient conditions on M M for B B to be right nonsingular and for A A to be the maximal right quotient ring of B B .

A module M is called dual automorphism invariant if whenever X1 and X2 are small submodules of M, then each epimorphism f : M/X1 → M/X2 lifts to an endomorphism g of M. A module M is said to be d-square free (dual square free) if whenever some factor module of M is isomorphic to N2 for a module N then N = 0. We show that each dual automorphism invariant module over a semiperfect ring which is a small epimorphic image of a projective lifting module is a direct sum of cyclic indecomposable d-square free modules. Moreover, we prove that for each module M over a semiperfect ring which is a small epimorphic image of a projective lifting module (e.g., M is a finitely generated module), M is dual automorphism invariant iff M is pseudoprojective. Also, we give the necessary and sufficient conditions for a dual automorphism invariant module over a right perfect ring to be quasiprojective.

Sufficient conditions are given, in module-theoretic terms, for the idealN(S) of the endomorphism ringS of a moduleM consisting of the endomorphisms with essential kernel to be nilpotent. This extends in a natural way several known results on the nilpotency ofN(S). WhenM is a quasi-injective module such thatS is right noetherian, it is shown thatS is right artinian if and only ifM has a finite rational Loewy series whose length is, in this case, equal to the index of nilpotency ofN(S).

Throughout this paper it is assumed that rings are associative, have the identity element, and all modules are left unital. R will denote a ring with identity, R-Mod the category of left R-modules, and for each left R-module M, E(M) (resp. J(M)) will represent the injective hull (resp. Jacobson radical) of M. Also, for a module M, A ⊆' M will mean that A is an essential submodule of M, and Z(M) denotes the singular submodule of M. M is called singular if Z(M) = M, and it is called non-singular in case Z(M) = 0. For fundamental definitions and results related to torsion theories, we refer to [12] and [14]. In this paper we shall deal mainly with Goldie torsion theory. Recall that a pair (G, F) of classes of left R-modules is known as Goldie torsion theory if G is the smallest torsion class containing all modules B/A, where A ⊆' B, and the torsion free class F is precisely the class of non-singular modules.