Semisimple Modules Research Articles

Commutative Rings whose Proper Ideals are Serial

A result of Nakayama and Skornyakov states that a ring R is an Artinian serial ring if and only if every R-module is serial. This motivated us to study commutative rings for which every proper ideal is serial. In this paper, we determine completely the structure of commutative rings R of which every proper ideal is serial. It is shown that every proper ideal of R is serial, if and only if, either R is a serial ring, or R is a local ring with maximal ideal ${\mathcal {M}}$ such that there exist a uniserial module U and a semisimple module T with ${\mathcal {M}}=U\oplus T$ . Moreover, in the latter case, every proper ideal of R is isomorphic to $U^{\prime }\oplus T^{\prime }$ , for some $U^{\prime }\leq U$ and $T^{\prime }\leq T$ . Furthermore, it is shown that every proper ideal of a commutative Noetherian ring R is serial, if and only if, either R is a finite direct product of discrete valuation domains and local Artinian principal ideal rings, or R is a local ring with maximal ideal ${\mathcal {M}}$ containing a set of elements {w 1,…,w n } such that ${\mathcal {M}}=\bigoplus _{i=1}^{n} Rw_{i}$ with at most one non-simple summand. Moreover, another equivalent condition states that: there exists an integer n ≥ 1 such that every proper ideal of R is a direct sum of at most n uniserial R-modules. Finally, we discuss some examples to illustrate our results.

Modules with epimorphisms on chains of submodules

We say that an [Formula: see text]-module [Formula: see text] satisfies epi-ACC on submodules if in every ascending chain of submodules of [Formula: see text], except probably a finite number, each module in chain is a homomorphic image of the next one. Noetherian modules, semisimple modules and Prüfer [Formula: see text]-groups have this property. Direct sums of modules with epi-ACC on submodules need not have this property. If [Formula: see text] satisfies epi-ACC on submodules, then [Formula: see text] is quasi-Frobenius. As a consequence, a ring [Formula: see text] in which all modules satisfy epi-ACC on submodules is an artinian principal ideal ring. Dually, we say that an [Formula: see text]-module [Formula: see text] satisfies epi-DCC on submodules if in every descending chain of submodules of [Formula: see text], except probably a finite number, each module in chain is a homomorphic image of the preceding. Artinian modules, semisimple modules and free modules over commutative principal ideal domains are examples of such modules. A semiprime right Goldie ring satisfies epi-DCC on right ideals if and only if it is a finite product of full matrix rings over principal right ideal domains. A ring [Formula: see text] for which all modules satisfy epi-DCC on submodules must be an artinian principal ideal ring.

Rings whose cyclic modules have restricted injectivity domains

A poor module is one that is injective relative only to semisimple modules and a module is maximally injective if its domain of injectivity is a coatom in the lattice of domains of injectivity (the so called injective profile of a ring). A ring is said to have no right middle class if every right module is injective or poor. We consider two families of rings characterized by their cyclic right modules: those whose nonzero cyclic modules are poor (property (P)) and those whose nonzero cyclic modules are maximally injective (property (Q)). We show that rings satisfying property (Q) are precisely those rings that have no right middle class and property (P). Structural properties of both classes of rings are obtained and it is shown, in particular, that a ring with property (Q) is isomorphic to a matrix ring over a nonsemisimple local right Artinian ring and is such that its right socle, right singular ideal and Jacobson radical coincide.

Irreducible components of varieties of representations: The local case

For any positive integer d, we determine the irreducible components of the varieties that parametrize the d-dimensional representations of a local truncated path algebra Λ. Here Λ is a quotient KQ/〈the paths of lengthL+1〉 of a path algebra KQ, where K is an algebraically closed field, L is a positive integer, and Q is the quiver with a single vertex and a finite number r of loops. The components are determined in both the classical and the Grassmannian settings, Repd(Λ) and GRASSd(Λ). Our method is to corner the components by way of a twin pair of upper semicontinuous maps from Repd(Λ) to a poset consisting of sequences of semisimple modules.An excerpt of the main result is as follows. Given a sequence S=(S0,…,SL) of semisimple modules with dim⁡⨁0≤l≤LSl=d, let RepS be the subvariety of Repd(Λ) consisting of the points that parametrize the modules with radical layering S. (The radical layering of a Λ-module M is the sequence (JlM/Jl+1M)0≤l≤L, where J is the Jacobson radical of Λ.) Suppose the quiver Q has r≥2 loops. If d≤L+1, the variety Repd(Λ) is irreducible and, generically, its modules are uniserial. If, on the other hand, d>L+1, then the irreducible components of Repd(Λ) are the closures of the subvarieties RepS for those sequences S which satisfy the inequalities dim⁡Sl≤r⋅dim⁡Sl+1 and dim⁡Sl+1≤r⋅dim⁡Sl for 0≤l<L; generically, the modules in any such component have socle layering (SL,…,S0). As a byproduct, the main result provides further installments of generic information on the modules corresponding to the irreducible components of the parametrizing varieties.

On generalized CS-modules

An S-closed submodule of a module M is a submodule N for which M/N is nonsingular. A module M is called a generalized CS-module (or briefly, GCS-module) if any S-closed submodule N of M is a direct summand of M. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right R-modules are projective if and only if all right R-modules are GCS-modules.

PSEUDO-RC-INJECTIVE MODULES

The main purpose of this work is to introduce and study the concept of pseudo-rc-injective module which is a proper generalization of rc-injective and pseudo-injective modules. Numerous properties and characterizations have been obtained. Some known results on pseudo-injective and rc-injective modules generalized to pseudo-rc-injective. Rationally extending modules and semisimple modules have been characterized in terms of pseud-rc-injective modules. We explain the relationships of pseudo-rc-injective with some notions such as Co-Hopfian , directly finite modules.

When is every linear transformation a sum of two commuting invertible ones?

A well-known result of Wolfson [7] and Zelinsky [8] says that every linear transformation of a vector space V over a division ring D is a sum of two invertible linear transformations except when dim(V)=1 and D=F2. Indeed, many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations. The goal of this note is to prove that every linear transformation of a vector space V over a division ring D is a sum of two commuting invertible ones if and only if |D|⩾3 and dim(V)<∞. As a consequence, a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms.

T-Semisimple Modules and T-Semisimple Rings

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z 2(M) ⊕ S(M) (where Z 2(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.

Comultiplication Modules over Noncommutative Rings

Comultiplication Modules over Noncommutative Rings

ON δ-LOCAL MODULES AND AMPLY δ-SUPPLEMENTED MODULES

The first part of this paper investigates the structure of δ-local modules. We prove that the following statements are equivalent for a module M: (i) M is δ-local; (ii) M is a coatomic module with either (a) M is a semisimple module having a maximal submodule N such that N is projective and M/N is singular, or (b) M has a unique essential maximal submodule K ≤ M such that for every maximal submodule L ≠ K, M/L is projective. The second part establishes some properties of finitely generated amply δ-supplemented modules.

On inclusion of permutation modules

Let H1, H2 be subgroups of a finite group G. Assume that G = ∪i=1mH2yiH1 = ∪j=1nH1gjH1 and that y1 = 1, g1 = 1. Let Di be the set consisting of right cosets of H2 contained in H2yiH1 and let dj (j = 1, ..., n) be the set consisting of right cosets contained in H1gjH{ia1}. We define the n×m matrix Mz (z = 1, ...,m) whose columns and rows are indexed by Di and dj respectively and the (dk,Dl) entry is \Dzgk ∩ Dl\. Let M = (M1, ..., Mm). Assume that \(1_{H_1 }^G\) and \(1_{H_2 }^G\) are semisimple permutation modules of a finite group G. In this paper, by using the matrix M, we give some sufficient and necessary conditions such that \(1_{H_1 }^G\) is isomorphic to a submodule of \(1_{H_2 }^G\). As an application, we prove Foulkes’ conjecture in special cases.

Socle chains of abelian regular semiartinian rings

Let R be an abelian regular and semiartinian ring with socle chain (Sα∣α≤σ). If λα denotes the rank of the semisimple module Sα+1/Sα, for every α<σ, then the dimension sequence(λα∣α<σ) is an invariant for R. By applying classical results of combinatorial set theory we prove necessary conditions satisfied by this invariant. On the other hand, we present constructions of commutative regular semiartinian rings with given ranks of slices of socle chain. In some particular cases we prove a necessary and sufficient condition under which there exists an abelian regular semiartinian ring with given ranks of slices.

Symmetric semisimple modules of group algebras over finite fields and self-dual permutation codes

A module of a finite group over a finite field with a symmetric non-degenerate bilinear form which is invariant by the group action is called a symmetric module. In this paper, a characterization of indecomposable orthogonal decompositions of symmetric semisimple modules and a criterion for the hyperbolic symmetric modules are obtained, and some applications to the self-dual permutation codes are shown.

Regular semiartinian rings

We study the structure of rings over which every right module is an essential extension of a semisimple module by an injective one. A ring R is called a right max-ring if every nonzero right R-module has a maximal submodule. We describe normal regular semiartinian rings whose endomorphism ring of the minimal injective cogenerator is a max-ring.

Prime–Extending Module and S-Prime Module

In this paper, the notation of prime -extending module, is introduced and studied where an R module M is Prime-extending if every nonzero proper submodule of M is essential in prime direct summand .Beside these, the notation of S-prime is given, where an R module is called S-prime if every proper direct summand is prime for this notation we proved: 1.Under the condition of S-prime module, M is Prime-extending if and only if M is extending .2. If M is semisimple module, then M is Prime-extending if and only if M is S-prime.

Rings whose modules have maximal or minimal projectivity domain

Using the notion of relative projectivity, projective modules may be thought of as being those which are projective relative to all others. In contrast, a module M is said to be projectively poor if it is projective relative only to semisimple modules. We prove that all rings have projectively poor modules. In fact, every ring even has a semisimple projectively poor module. Properties of projectively poor modules are studied and particular emphasis is given to the study of modules over PCI domains; we note that over such domains when all right ideals are principal most modules seem to be either projective or projectively poor. We consider rings over which modules are either projective or projectively poor and call them rings without a p-middle class. We show that a QF ring R with homogeneous right socle and J ( R ) 2 = 0 has no right p-middle class. As we analyze the structure of rings with no right p-middle class, among other results, we show that any such ring is the ring direct sum of a semisimple artinian ring and a ring K which is either zero or an indecomposable ring such that either (i) K is a semiprimary right SI-ring with J ( K ) ≠ 0 , or (ii) K is a semiprimary ring with S o c ( K K ) = Z r ( K ) = J ( K ) ≠ 0 , or (iii) K is a prime ring with S o c ( K K ) = 0 , and either J ( K ) = 0 or J K ( K ) and J ( K ) K are infinitely generated, or (iv) K is a prime right SI-ring with infinitely generated right socle.

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.

MODULES WITH COINDEPENDENT MAXIMAL SUBMODULES

Let R be a ring with identity. A unital left R-module M has the min-property provided the simple submodules of M are independent. On the other hand a left R-module M has the complete max-property provided the maximal submodules of M are completely coindependent, in other words every maximal submodule of M does not contain the intersection of the other maximal submodules of M. A semisimple module X has the min-property if and only if X does not contain distinct isomorphic simple submodules and this occurs if and only if X has the complete max-property. A left R-module M has the max-property if [Formula: see text] for every positive integer n and distinct maximal submodules L, Li(1 ≤ i ≤ n) of M. It is proved that a left R-module M has the complete max-property if and only if M has the max-property and every maximal submodule of M/Rad M is a direct summand, where Rad M denotes the radical of M, and in this case every maximal submodule of M is fully invariant. Various characterizations are given for when a module M has the max-property and when M has the complete max-property.

Rings whose modules have maximal or minimal injectivity domains

In a recent paper, Alahmadi, Alkan and López–Permouth defined a module M to be poor if M is injective relative only to semisimple modules, and a ring to have no right middle class if every right module is poor or injective. We prove that every ring has a poor module, and characterize rings with semisimple poor modules. Next, a ring with no right middle class is proved to be the ring direct sum of a semisimple Artinian ring and a ring T which is either zero or of one of the following types: (i) Morita equivalent to a right PCI-domain, (ii) an indecomposable right SI-ring which is either right Artinian or a right V-ring, and such that soc ( T T ) is homogeneous and essential in T T and T has a unique simple singular right module, or (iii) an indecomposable right Artinian ring with homogeneous right socle coinciding with the Jacobson radical and the right singular ideal, and with unique non-injective simple right module. In case (iii) either T T is poor or T is a QF-ring with J ( T ) 2 = 0 . Converses of these cases are discussed. It is shown, in particular, that a QF-ring R with J ( R ) 2 = 0 and homogeneous right socle has no middle class.

Presenting degenerate Ringel–Hall algebras of cyclic quivers

Using the generators labelled by simple and sincere semisimple modules for the Ringel–Hall algebra H q ( n ) of a cyclic quiver Δ ( n ) , we give a presentation for the degenerate algebra H 0 ( n ) . This is achieved by establishing a presentation for the generic extension monoid algebra of Δ ( n ) . As an application, we show that both the degenerate Ringel–Hall algebra and the degenerate quantum affine s l n admit multiplicative bases.