Direct Summand Research Articles

Property (Baw1) and Weyl type theorems

In this paper, we study property (Baw1) for bounded linear operators defined by Zariouh and Lombarkia in [12] as a variant of Weyl type theorem. We discuss property (Baw1) for operators satisfying single valued extension property (SVEP). We explore conditions related to the direct summands to ensure the extension of the property (Baw1) from and to their direct sum .

A strong Differentiable Absorption of Hilbert 𝑪∗-Modules with Connections, and Lifts of Unbounded Operators

The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert 𝐶∗-module is isomorphic to a direct summand in the standard module of square summable sequences in the base 𝐶∗-algebra. This result be generalized byJens Kaad [Jka10] by incorporating a densely defined derivation on the base 𝐶∗-algebra. It following the perfect densely method of Jens Kaad[Jka10] leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The differentiable absorption theorem is then applied to construct densely defined connections (or correspondences) on Hilbert 𝐶∗-modules. These connections can in turn be used to define selfadjoint and regular ”lifts” of unbounded operators which act on an auxiliary Hilbert 𝐶∗- module.

ON FS-LIFTING AND CFS-LIFTING SEMIMODULES OVER SEMIRINGS

In this paper, we introduce the notions of fs-lifting and cfs-lifting semimodules as respectively the generalizations of finitely lifting (f-lifting for short) semimodules and cofinitely lifting (cf-lifting) semimodules. Under some conditions, we prove some results on fs-lifting and cfs-lifting semimodules for proving the equivalence between fs-lifting and cfs-lifting semimodules. An $R$-semimodule $M$ is fs-lifting if every finitely generated subtractive subsemimodule $N$ of $M$, there exists a direct summand $K$ of $M$ such that $K\leq N$ and $N/K\ll M/K$; so if every coessential finitely generated subtractive subsemimodule $N$ of $M$, there exists a direct summand $K$ of $M$ such that $K\leq N$ and $N/K\ll M/K$, $M$ is called cfs-lifting.

Semi-generalized Bassian groups

Abstract As a common non-trivial generalization of the concept of a (proper) generalized Bassian group, we introduce the notion of a semi-generalized Bassian group and initiate its comprehensive investigation. Precisely, we give a satisfactory characterization of these groups by showing in the cases of 𝑝-torsion groups, torsion-free groups and splitting mixed groups their complete description. Moreover, we classify the groups with the clearly related property that every subgroup is essential in a direct summand of the whole group.

Extending modules and generalizations via left exact preradicals

A module is said to be extending if every submodule is essential in a direct summand of the module. In this paper, we define and investigate in details extending notion and its two generalizations in terms of a left exact preradical in the category of right modules. Our results are based on structural and decomposition properties of these new classes of modules. Examples are included on using special left exact preradicals.

Big pure projective modules over commutative noetherian rings: Comparison with the completion

Abstract A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add ⁡ ( M ) {\operatorname{Add}(M)} , which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add ⁡ ( M ) {\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V * ⁢ ( M ) {V^{*}(M)} , of isomorphism classes of countably generated modules in Add ⁡ ( M ) {\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V * ⁢ ( M ) {V^{*}(M)} is a submonoid of V * ⁢ ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} , this allows us to make computations with examples and to prove some realization results.

On modules in which every proper finitely generated submodule is a subset of a kernel of a nonzero endomorphism

We study an R-module M such that for every proper finitely generated submodule K of M, Hom R ( M / K , M ) ≠ 0 . Such modules are called co-finite-retractable (simply CFR). We consider when factor modules, direct sums and direct summands of CFR modules are CFR. Also, we investigate the injective hull of CFR modules. We explore certain properties of a ring R such that every (cyclic, free) module is CFR. Also, we show that R is a semisimple ring if and only if every R-module is regular if and only if every CFR module over R is regular.

Simple-separable modules

A module $M$ over a ring is called simple-separable if every simple submodule of $M$ is contained in a finitely generated direct summand of $M$. While a direct sum of any family of simple-separable modules is shown to be always simple-separable, we prove that a direct summand of a simple-separable module does not inherit the property, in general. It is also shown that an injective module $M$ over a right noetherian ring is simple-separable if and only if $M=M_1 \oplus M_2$ such that $M_1$ is separable and $M_2$ has zero socle. The structure of simple-separable abelian groups is completely described.

A Study On a New Generalization of $\delta$-Supplemented Modules

For any ring $S$ and an $S$-module $W$, a submodule $G$ of $W$ is termed \emph{co$_\delta$-coatomic} if the quotient module $W/G$ is $\delta$-coatomic. In this study, we introduce the term ($\oplus$-)\emph{co$_\delta$-coatomically $\delta$-supplemented module}, or shortly ($\oplus$-)\emph{co$_\delta$-$\delta$-supplemented module} to describe a module $W$ where each co$_\delta$-coatomic submodule has a $\delta$-supplement (that is a direct summand) in $W$. Furthermore, a module $W$ is identified as \emph{co$_\delta$-coatomically $\delta$-semiperfect}, or shortly \emph{co$_\delta$-$\delta$-semiperfect}, provided each $\delta$-coatomic quotient module of $W$ has a projective $\delta$-cover. It has been proved that over a $\delta$-semiperfect ring $S$, the module $_{S}S$ is $\oplus_{\delta}$-co-coatomically supplemented if and only if $_{S}S$ is co$_\delta$-$\delta$-semiperfect if and only if $_{S}S$ is $\oplus$-co$_\delta$-$\delta$-supplemented.

On polynomial invariant rings in modular invariant theory

Let k be a field of characteristic p>0, V a finite-dimensional k-vector-space, and G a finite p-group acting k-linearly on V. Let S=SymV⁎. Confirming a conjecture of Shank-Wehlau-Broer, we show that if SG is a direct summand of S, then SG is a polynomial ring, in the following cases:(a)k=Fp and dimk⁡V=4; or(b)|G|=p3. In order to prove the above result, we also show that if dimk⁡VG≥dimk⁡V−2, then the Hilbert ideal hG,S is a complete intersection.

〖FI〗_ss-Lifting Modules

The purpose of this note is to show some key features of 〖FI〗_ss-lifting and strongly 〖FI〗_ss-lifting modules. We examine that under whose condition for direct summands, direct sums and submodules of (strongly) 〖FI〗_ss-lifting modules are (strongly) 〖FI〗_ss-lifting. We give an example to exhibit that an FI-lifting module needs not to be 〖FI〗_ss-lifting. We provide that the property of being strongly 〖FI〗_ss-lifting module is inherited by direct summands.

When some complement of an exact fully invariant submodule is a direct summand

In this paper, we study modules with the property that every exact fully invariant submodule has a complement which is a direct summand. This new class of modules properly contains both of classes of FI-extending modules and [Formula: see text]-modules (i.e. modules have [Formula: see text] property for exact submodules). We obtain several module theoretical results including decompositions for the former class of modules.

Quasi-hereditary covers of Temperley–Lieb algebras and relative dominant dimension

Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras $S(2, d)$ and their $q$ -analogues. Our second main result completely determines the relative dominant dimension of $S(2, d)$ with respect to $Q=V^{\otimes d}$ , the $d$ -th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of $q$ -Schur algebras $S(2,d)$ give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of the $q$ -Schur algebra $S(2, d)$ and precisely how far these two algebras are from being Morita equivalent, when they are not. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$ -Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley–Lieb algebra.

AN APPROACH FOR δ_SS-SUPPLEMENTED MODULES WITH IDEALS

The aim of this paper is to present J_(δ_ss-)supplemented modules and investigate their main algebraic properties. Let J be an ideal of a ring S and A be an S-module. We call a module A is J_(δ_ss )-supplemented, provided for each submodule B of A, there exists a direct summand C of A such that A=B+C, B∩C≤JC and B∩C≤〖Soc〗_δ (C). We prove that the factor module by any fully invariant submodule remains so, when the module is J_(δ_ss )-supplemented. We show that any direct sum of J_(δ_ss )-supplemented modules preserves its J_(δ_ss )-supplemented property when this direct sum is a duo module. Additionally, we make comparisons of J_(δ_ss )-supplemented modules with other module types.

On tau-discrete modules

An R-module M is said to be (quasi) τ-discrete if M is τ-lifting and has the property (D_2) (respectively, has the property (D_3)), where τ is a preradical in R-mod. It is shown that: (1) direct summands of a (quasi) τ-discrete module are (quasi) τ-discrete; (2) a projective module M is τ-discrete iff M/(τ(M)) is semisimple and τ(M) is QSL; (3) if a projective module M is Soc-lifting, then M/(Soc(M)) is Soc-discrete and Rad(M/Soc(M) ) is semisimple.

A Research on the Generalizations of Modules Whose Submodules are Isomorphic to a Direct Summand

A module M is called virtually semisimple (resp. virtually extending) if every submodule (resp. complement submodule) of M is isomorphic to a direct summand of M. It is known that virtually extending modules is a generalization of virtually semisimple modules. In this paper, the relationships between virtually extending modules and other generalizations of virtually semisimple modules are examined. Moreover, we introduce a new generalization of virtually semisimple modules; namely CH modules: We say a module M is a c-epi-retractable (or briefly CH module) if any complement submodule of M is a homomorphic image of M. CH modules contains the class of virtually extending modules and the class of epi-retractable modules. We also give some basic properties of this new module class.

Rings Characterizations with Mutually SS-Supplemented Modules

In this text, the notion of mutually ss-supplemented modules characterizes with the help of semiperfect rings. For this mutually ss-supplemented modules were first classified according to certain properties. These features can be listed as follows in the refinable modules, distributive modules, fully invariant submodules and (π-)projective modules. Especially it was proven that each 𝜋-projective mutually ss-supplemented module is amply mutually ss-supplemented. It was obtained that every ss-supplemented refinable module has direct summand which is a mutually ss-supplement in the module. It was shown that C=⨁_(ϱ∈Λ ) C_ϱ is a mutually ss-supplemented module which each submodule of C is a fully invariant submodule, for the family of mutually ss-supplemented modules {C_ϱ }_(ϱ∈Λ).

GRAPH CHARACTERISATION OF THE ANNIHILATOR IDEALS OF LEAVITT PATH ALGEBRAS

Abstract If E is a graph and K is a field, we consider an ideal I of the Leavitt path algebra $L_K(E)$ of E over K. We describe the admissible pair corresponding to the smallest graded ideal which contains I where the grading in question is the natural grading of $L_K(E)$ by ${\mathbb {Z}}$ . Using this description, we show that the right and the left annihilators of I are equal (which may be somewhat surprising given that I may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterisation from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on E which is equivalent to unital $L_K(E)$ having this property.

Direct Summand of Serial Modules

Let R be an associative ring and M a unitary left R-module. An R-module M is said to be uniserial if its submodules are linearly ordered by inclusion. A serial module is a direct sum of uniserial modules. In this paper, we bring our modest contribution to the open problem listed in the book of Alberto Facchini "Module Theory" which states that is any direct summand of a serial module serial? The answer is yes for particular rings and R-modules.

Examples of compact quantum groups with [formula omitted] a factor

For each λ∈]0,1] we exhibit an uncountable family of compact quantum groups G such that the von Neumann algebra L∞(G) is the injective factor of type IIIλ with separable predual. We also show that uncountably many injective factors of type III0 arise as L∞(G) for some compact quantum group G. We introduce natural invariants of quantum groups related to the scaling group, modeled on the Connes invariant T for von Neumann algebras. We compute the values of these invariants in our examples, which allows us to distinguish between them. We also investigate their connection to the Connes invariants T(L∞(G)), S(L∞(G)).In the final section we show that for a compact quantum group G the von Neumann algebra L∞(G) cannot have a direct summand of the form B(H) with dim⁡H=∞. In particular, factors of type I (of any dimension) cannot be obtained as L∞(G) for a non-trivial compact quantum group G.