Faithful Module Research Articles

Classification of multiplication modules over multiplication rings with finitely many minimal primes

Abstract A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterization of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive modules. It is proven that for a multiplication ring with finitely many minimal primes every faithful, Noetherian, distributive module is a faithful multiplication module, and vice versa.

Right ideal decompositions of nearrings with faithful 2-step modules

When N is a left nearring with identity satisfying the descending chain condition on right ideals N can be written as a direct sum of nonzero right ideals that cannot be further decomposed into direct sums of nonzero right ideals. If N is a ring the nature of this decomposition is well understood. A nearring module of N possessing a unique proper nonzero N-subgroup is called a 2-step module. Here we initiate the study of the nature of the previously described decompositions when N is not a ring by considering the situation where N has a faithful 2-step module.

Strongly nearly-2-absorbing submodules

Let Ɍ be a commutative ring with identity and V be a unitary left Ɍ-module. The concept of Strongly Nearly-2-Absorabing sub-modules as a generalization of Endo-2-Absorabing sub-modules and strong form of Nearly-2-Absorabing sub-modules are introduced in this paper. Many examples, basic properties of this concept are introduced. Furthermore we prove that in class of (scalar, cyclic and finitely-generated) modules the two concepts Nearly-2-Absorabing sub-modules and Strongly Nearly-2-Absorabing sub-modules are equivalent. Moreover we prove that Endo-2-Absorabing and Strongly Nearly-2-Absorabing sub-modules are equivalent in class of (semi simple, regular) modules. Also those concepts are equivalent in class of all modules over a v-ring. Finally we prove several characterizations of Strongly Nearly-2-Absorabing sub-modules in some types of modules such as (projective, faithful and content) modules in class of cyclic modules.

Fuzzy generator and fuzzy cogenerator

In this article, we introduce the fuzzy generator and fuzzy cogenerator and investigate some of their properties. Necessary and sufficient conditions for a fuzzy module to become a fuzzy generator and sufficient conditions for a fuzzy module to become a fuzzy cogenerator have also been established. We also discuss the behavior of fuzzy modules of finitely generated modules. Further, we introduce fuzzy trace and fuzzy reject and establish some specific results. Finally, we define a faithful fuzzy module and weave this notion with the concepts of fuzzy generator and fuzzy cogenerator.

Multiplication Module and Retractable Concept

In this paper; we study a strong relation between faithful multiplication module and retractable concept. We proved that if M is faithful and cyclic, so is retractable. On the other hand, faithful injective Module with multiplication property gives retractable module. Also, finitely generated semi-simple module over quasi-invariant ring is retractable module. Finally, some definitions, examples and properties have been presented about retractable module.

Strongly Pseudo Nearly Semei-2-Absorbing Submodules (II)

Let be a module over a commutative ring with identity. Before studying the concept of the Strongly Pseudo Nearly Semi-2-Absorbing submodule, we need to mention the ideal and the basics that you need to study the concept of the Strongly Pseudo Nearly Semi-2-Absorbing submodule. Also, we introduce several characteristics of the Strongly Pseudo Nearly Semi-2-Absorbing submodule in classes of multiplication modules and other types of modules. We also had no luck because the ideal is not a Strongly Pseudo Nearly Semi-2-Absorbing ideal. Also, it is noted that is the Strongly Pseudo Nearly Semi-2-Absorbing ideal under several conditions, which is this faithful module, projective module, Z-regular module and content module and non-singular module and content module. Also we introduce the characterization of the concept of the Strongly Pseudo Nearly Semi-2-Absorbing ideals by a special kind of submodule

On the structure of groups admitting faithful modules with certain conditions of primitivity

In the paper we study structure of soluble-by-finite groups of finite torsion-free rank which admit faithful modules with conditions of primitivity. In particular, we prove that under some additional conditions if an infinite finitely generated linear group $G$ of finite rank admits a fully primitive fully faithful module then $G$ has infinite $FC$-centre.

Fuzzy visible submodules in the class of fuzzy faithful multiplication modules

In this paper, the behavior of fuzzy visible submodules and its effect of it in the class of fuzzy faithful multiplication modules have been studied. Some relationship between fuzzy visible modules and fuzzy residual quotient ideals has been presented. Also, other properties with important results have been studied.

Quasi $J$-submodules

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. The aim of this paper is to extend the notion of quasi $J$-ideals of commutative rings to quasi $J$-submodules of modules. We call a proper submodule $N$ of $M$ a quasi $J$-submodule if whenever $r\in R$ and $m\in M$ such that $rm\in N$ and $r\notin(J(R)M:M)$, then $m\in M$-$rad(N)$. We present various properties and characterizations of this concept (especially in finitely generated faithful multiplication modules). Furthermore, we provide new classes of modules generalizing presimplifiable modules and justify their relation with (quasi) $J$-submodules. Finally, for a submodule $N$ of $M$ and an ideal $I$ of $R$, we characterize the quasi $J$-ideals of the idealization ring $R(+)M$.

Characterizations of Weakly Approximately Primary Submodules in Some Types of Modules

Our aim in this note is to introduce several characterizations of weakly approximately primary submodules in class of multiplication modules. Furthermore, we characterized weakly approximately primary submodules by theirs resudule in the class of multiplication modules with the help of some types of modules as non-singular, projective, regular and faithful modules. Also, we characterized weakly approximately primary ideal with some kind of weakly approximately primary submodules in the same previous classes of modules with help of finitely generated modules.

A Characterisation of Morita Algebras in Terms of Covers

A pair (A, P) is called a cover of EndA(P)op if the Schur functor HomA(P,−) is fully faithful on the full subcategory of projective A-modules, for a given projective A-module P. By definition, Morita algebras are the covers of self-injective algebras and then P is a faithful projective-injective module. Conversely, we show that A is a Morita algebra and EndA(P)op is self-injective whenever (A, P) is a cover of EndA(P)op for a faithful projective-injective module P.

Characterizations of certain Morita context rings

Characterizations of Morita context rings satisfying various additional ring properties, including reflexive, semiprime, prime, NI, weakly 2-primal and 2-primal, are given. The connections between the characterizations of reflexive, semiprime and prime Morita context rings are outlined and shown to be related to torsionless and faithful modules. Similarly, the characterizations of NI, weakly 2-primal and 2-primal are shown to be related. In particular, a characterization of NI Morita context rings is determined to be equivalent to Köthe’s conjecture.

Finite singular orbit modules for algebraic groups

Building on the classification of modules for algebraic groups with finitely many orbits on subspaces [9], we determine all faithful irreducible modules V for a connected simple algebraic group H, such that H≤SO(V) and H has finitely many orbits on singular 1-spaces of V. We do the same for H connected semisimple, and maximal among connected semisimple subgroups. This question is naturally connected with the problem of finding for which pairs of subgroups H,K of an algebraic group G there are finitely many (H,K)-double cosets. This paper provides a solution to the question when K is a maximal parabolic subgroup P1 of a classical group SOn, where P1 is the stabilizer of a singular 1-space. We find an interesting range of new examples, from a 5-dimensional module for A1 in characteristic p>5, to the spin module for B6 when p=2.

Modules with annihilation property

Let [Formula: see text] be an associative ring with identity. A right [Formula: see text]-module [Formula: see text] is said to have Property ([Formula: see text]), if each finitely generated ideal [Formula: see text] has a nonzero annihilator in [Formula: see text]. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ([Formula: see text]). We study and construct various classes of modules with Property ([Formula: see text]). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce [Formula: see text]-dual McCoy modules and show that, for every strictly totally ordered monoid [Formula: see text], faithful symmetric modules are [Formula: see text]-dual McCoy. We then use this notion to give a characterization for modules with Property ([Formula: see text]). For a faithful symmetric right [Formula: see text]-module [Formula: see text] and a strictly totally ordered monoid [Formula: see text], it is proved that the right [Formula: see text]-module [Formula: see text] is primal if and only if [Formula: see text] is primal with Property ([Formula: see text]).

On faithfully balanced modules, F-cotilting and F-Auslander algebras

The aim of this paper is to give a unification of the theory of (co)tilting modules and (higher) Auslander correspondence by using faithfully balanced modules (i.e., faithful modules with the double centralizer property). In the language of relative homological algebra following Auslander-Solberg, a relative version of faithfully balancedness is given, which will be called 1-F-faithful. The relative versions of (higher) Auslander correspondence and Brenner-Butler theorem are also developed.

Residually faithful modules and the Cohen–Macaulay type of idealizations

The Cohen–Macaulay type of idealizations of maximal Cohen–Macaulay modules over Cohen–Macaulay local rings is closely explored. There are two extremal cases, one of which is related to the theory of Ulrich modules, and the other one is related to the theory of residually faithful modules and closed ideals, developed by Brennan and Vasconcelos.

Reflexive ideals and completely faithful modules of Iwasawa algebras

Let p≥5 be a prime number. Let G=H×Z, where H is a torsion free compact p-adic analytic group such that its Lie algebra is split semisimple over Qp and Z≅Znp, where n∈N0. We classify all the prime c-ideals of the Iwasawa algebra ΛG. Then we show the following: Let M be a finitely generated torsion ΛG-module, such that it has no non-zero pseudo-null submodules. Let q(M) denote the image of M in the quotient category mod-ΛG/CΛG1 via the quotient functor q, where CΛG1 denotes the Serre-subcategory of pseudo-null ΛG-modules in mod-ΛG. Then q(M) is completely faithful if and only if M is ΛZ-torsion free.

Monomial algebras of dominant dimension at least two

Let A be a finite dimensional algebra having the double centralizer property with respect to a minimal faithful projective-injective left module Af for some idempotent f. We prove that in this case A is a monomial algebra if and only if A is a Nakayama algebra given by a quiver with admissible relations.

Complex Lipschitz structures and bundles of complex Clifford modules

Let (M,g) be a pseudo-Riemannian manifold of signature (p,q). We construct mutually quasi-inverse equivalences between the groupoid of bundles of weakly-faithful complex Clifford modules on (M,g) and the groupoid of reduced complex Lipschitz structures on (M,g). As an application, we show that (M,g) admits a bundle of irreducible complex Clifford modules if and only if it admits either a Spinc(p,q) structure (when p+q is odd) or a Pinc(p,q) structure (when p+q is even). When p−q≡83,4,6,7, we compare with the classification of bundles of irreducible real Clifford modules which we obtained in previous work. The results obtained in this note form a counterpart of the classification of bundles of faithful complex Clifford modules which was previously given by T. Friedrich and A. Trautman.

Generators of split extensions of Abelian groups by cyclic groups

Let G \simeq M \rtimes C be an n -generator group which is a split extension of an Abelian group M by a cyclic group C . We study the Nielsen equivalence classes and T-systems of generating n -tuples of G . The subgroup M can be turned into a finitely generated faithful module over a suitable quotient R of the integral group ring of C . When C is infinite, we show that the Nielsen equivalence classes of the generating n -tuples of G correspond bijectively to the orbits of unimodular rows in M^{n -1} under the action of a subgroup of GL _{n - 1}(R) . Making no assumption on the cardinality of C , we exhibit a complete invariant of Nielsen equivalence in the case M \simeq R . As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag–Solitar groups, split metacyclic groups and lamplighter groups.