Nonzero Submodules Research Articles

Semi-Essentially Compressible Modules and Semi-Essentially Retractable Modules

Let be a commutative ring with 1 and be a left unitary . In this paper, the generalizations for the notions of compressible module and retractable module are given. An is termed to be semi-essentially compressible if can be embedded in every of a non-zero semi-essential submodules. An is termed a semi-essentially retractable module, if for every non-zero semi-essentially submodule of an . Some of their advantages characterizations and examples are given. We also study the relation between these classes and some other classes of modules.

Semi-essentially prime modules

Let R be a commutative ring with 1 and M be a (left) unitary R – module. This essay gives generalizations for the notions prime module and some concepts related to it. We termed an R – module M as semi-essentially prime if annR (M) = annR (N) for every non-zero semi-essential submodules N of M. Given some of their advantages characterizations and examples, and we study the relation between these and some classes of modules.

On atomic ideals in some factor rings of $C(X,\Bbb Z)$

A nonzero $R$-module $M$ is atomic if for each two nonzero elements $a, b$ in $M$, both cyclic submodules $Ra$ and $Rb$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$-space $X$, the factor rings $C(X,\Bbb{Z})/C_F(X,\Bbb{Z})$ and $C_c(X)/C_F(X)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$, the factor ring $\Bbb{Z}^X/ \Bbb{Z}^{(X)}$ has no atomic ideal. Another result is that for each infinite $P$-space $X$, the socle of the factor ring $C_c(X)/C_F(X)$ is always equal to zero. Also, zero-dimensional spaces $X$ are characterized for which $C^F(X,\Bbb{Z})/C_F(X,\Bbb{Z})$ have atomic ideals.

Perpendicular graph of modules

Let $R$ be a ring and $M$ be an $R$-module. Two modules $A$ and $B$ are called orthogonal, written $A\perp B$, if they do not have non-zero isomorphic submodules. We associate a graph $\Gamma_{\bot}(M)$ to $M$ with vertices $\mathcal{M}_{\perp}=\{(0)\neq A\leq M \mid \exists B\neq (0) \;\mbox{such that}\; A\perp B\}$, and for distinct $A,B\in \mathcal{M}_{\perp}$, the vertices $A$ and $B$ are adjacent if and only if $A\perp B$. The main object of this article is to study the interplay of module-theoretic properties of $M$ with graph-theoretic properties of $\Gamma_{\bot}(M)$. An algorithm is given to generate perpendicular graphs of $\mathbb{Z}_n$.

Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

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.

Iso-retractable modules and rings

The modules which are isomorphic to their non-zero submodules are known as iso-retractable. We characterize simple modules in terms of iso-retractable modules. We provide several sufficient conditions for iso-retractable modules to be simple. We show that if the endomorphism ring of an iso-retractable module is von-Neumann regular then [Formula: see text] is a simple module. In general, iso-retractable modules need not be projective (injective) and vice versa. We investigate some properties of iso-retractable modules with projectivity as well as injectivity. Finally, we provide some open problems.

Pseudo Projective Modules Which are not Quasi Projective and Quivers

In this paper we construct pseudo projective modules which are not quasi projective over non-commutative perfect rings. To do it we construct finite dimensional quiver algebras over the field $\mathbb{Z}_2$. The modules which are constructed will have finite length three and only three nonzero proper submodules.

The total graph of unfaithful submodules of a module over reversible rings

Let [Formula: see text] be an associative ring with identity. A ring [Formula: see text] is called reversible if [Formula: see text], then [Formula: see text] for [Formula: see text]. The total graph of unfaithful submodules of a module [Formula: see text] over a reversible ring [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are all nonzero unfaithful submodules of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] is unfaithful. In this paper, we determine the diameter and girth of [Formula: see text]. Also, we study some combinatorial properties of [Formula: see text] such as independence number and clique number. Moreover, we study the case that the degree of a vertex of [Formula: see text] is finite.

A characterization of modules embedding in products of primes and enveloping condition for their class

Modules (cogenerated by nonzero their submodules) having nonzero square homomorphism in nonzero submodules are said (prime) weakly compressible (wc). Such modules are semiprime (i.e. they are cogenerated by their essential submodules). For many rings R, including commutative rings, it is proved that wc modules are isomorphic submodules of products of prime modules, and the converse holds precisely when the class of wc modules is enveloping for mod-R or equivalently, every semiprime module is wc. Semi-Artinian rings R have the latter property and the converse is true when R is strongly regular. Duo Noetherian rings over which wc modules form an enveloping class are shown to have a finite number maximal ideals. If R is Morita equivalent to a Dedekind domain, then the class of wc R-modules is enveloping if and only if R is simple Artinian or J (R) ≠ 0.

Comultiplication Modules over Noncommutative Rings

Comultiplication Modules over Noncommutative Rings

A note on Selmer groups of abelian varieties over the trivializing extensions

We prove that for any abelian variety A A defined over a number field K K that is not isogenous to a product of CM elliptic curves, the pontrjagin dual of the Selmer group of the abelian variety over the trivializing extension K ( A [ p ∞ ] ) K(A[p^\infty ]) has no nonzero pseudo-null submodules.

Type Submodules and Direct Sum Decompositions of Modules

A type decomposition of a module M over a ring R is a direct sum decomposition for which any two distinct summands have no nonzero isomorphic submodules. In this paper, we investigate when a module possesses certain kinds of type decompositions and when such decompositions are unique. Introduction. It is well known that every torsion abelian group has a unique decomposition into its p-torsion subgroups. By GoodearlBoyle [4], every nonsingular injective module E has a unique decomposition E = E1 ⊕ E2 ⊕ E3 where E1, E2, E3 are of types I, II, III respectively, see Definition 2.7. Why do such decompositions exist? Why are such decompositions unique? Are there any common things between these two results? All these questions will be answered in this paper. In fact, we can present a more general theory on existence and uniqueness of type decompositions of modules, so that the above results, as well as many other known results, are obtained as very special cases. The common property for certain diverse kinds of direct sum decompositions of modules M including the two decompositions above is that any two distinct direct summands have no nonzero isomorphic submodules, or equivalently all direct summands are what we will call type submodules. The cause for the existence of such decompositions is that these modules M have a ‘decomposability property’ which will be discussed in detail in Section 1, while the uniqueness of such direct sum decompositions is ensured by a module property called UTC. A theory of such modules is developed in Section 2. Throughout, all rings R are associative with identity and modules are unital right R-modules and M is an R-module. A class K of modules is a type, or natural class, if it is closed under isomorphic copies, 1991 AMS Mathematics Subject Classification. Primary 16D70, 16D80. Research of the second author was supported by NSERC Grant OGP0194196. Copyright c ©2005 Rocky Mountain Mathematics Consortium

On τ-completely decomposable modules

For a hereditary torsion theory τ, a moduleAis called τ-completedly decomposable if it is a direct sum of modules that are the τ-injective hull of each of their non-zero submodules. We give a positive answer in several cases to the following generalised Matlis' problem: Is every direct summand of a τ-completely decomposable module still τ-completely decomposable? Secondly, for a commutative Noetherian ringRthat is not a domain, we determine those torsion theories with the property that every τ-injective module is an essential extension of a (τ-injective) τ-completely decomposable module.

Rings Whose Nonzero Modules Have Maximal Submodules

All rings are assumed to be associative and (except for nil-rings and some stipulated cases) to have nonzero identity elements. Expressions such as a “Noetherian ring” mean that the corresponding right and left conditions hold. A module is said to be simple if it does not have nonzero proper submodules. A submodule N of a module M is called a maximal submodule (in M) if the factor module M/N is simple. A ring A is called a right max ring (or a right Bass ring) if every nonzero right A-module has a maximal submodule. A ring is said to be orthogonally finite if it does not contain an infinite set of nonzero orthogonal idempotents. Bass [11] proved that a ring with the minimum condition on principal left ideals is an orthogonally finite right max ring. Cozzens [25] and Koifman [61] constructed examples of orthogonally finite right max rings which do not satisfy the minimum condition on principal left ideals. This review contains some new and some old results related to max rings. We present the necessary notation and definitions. Let A be a ring. We denote the center of the ring A by C(A). A subset B of A is said to be central (in A) if B ⊆ C(A). If M and N are two modules, then Hom(M,N) denotes the Abelian group formed by all homomorphisms M → N . For a module M , we denote by End(M) the endomorphism ring Hom(M,M) of M . For a subset B of A, the right annihilator and the left annihilator of B in A are denoted by rA(B) and A(B), respectively. We can omit the subscripts if the situation is obvious. An element a of A is said to be right regular (resp. left regular) in A if r(a) = 0 (resp. (a) = 0). A ring A is said to be regular if for every element a of A, there exists an element b of A such that a = aba. A ring A is said to be right weakly regular if B2 = B for every right ideal B of A. A ring without nonzero nilpotent ideals is called a semiprime ring. A ring is called a prime ring if the product of any two of its nonzero ideals is not equal to zero. A ring A is said to be simple if each of its nonzero ideals coincides with A. A ring is called a domain if the product of any two of its nonzero elements is not equal to zero. For a ring A, an ideal P of A is called a prime (resp. semiprime and completely prime) ideal if the factor ring A/P is a prime ring (resp. a semiprime ring and a domain). A ring A is called a P.I.-ring (or a ring with polynomial identity) if A satisfies the polynomial identity f(x1, . . . , xn) = 0, where f(x1, . . . , xn) is a polynomial in noncommutative variables with coefficients in the ring of integers Z, and Z coincides with its ideal generated by the coefficients of f(x1, . . . , xn). A submodule N of a module M is said to be essential (in M) if N has nonzero intersection with any nonzero submodule of the module M . In this case, we say that M is an essential extension of the module N . A submodule N of a module M is called a superfluous submodule (in M) if N +M ′ = M for every proper submodule M ′ of M .

Forcing Linearity Numbers for Modules over PID's

Let V be a module over a principal ideal domain. Then V = M ⊕ N where M is divisible and N has no nonzero divisible submodules. In this paper we determine the forcing linearity number for V when N is a direct sum of cyclic modules. As a consequence, the forcing linearity numbers of several classes of Abelian groups are obtained.

Injective resolutions of some regular rings

Let A be a noetherian Auslander regular ring and δ the canonical dimension function on A-modules, which is defined as δ(M)=d−j(M) where d is the global dimension of A and j(M) is the grade of M. An A-module is s- pure if δ(N)=s for all its non-zero noetherian submodules N, and is essentially s- pure if it contains an essential submodule which is s-pure. Consider a minimal injective resolution of A as an A-module 0→A→I 0→I 1→⋯→I d→0. We say A has a pure (resp. essentially pure) injective resolution if I i is (d−i)-pure (resp. essentially (d−i)-pure). We show that several classes of Auslander regular rings with global dimension at most 4 have pure or essentially pure injective resolutions.

On Infinite Goldie Dimension

Two elementsxandyof a partially ordered setPare said to bedisjointif there is noz∈Psuch thatz≤xandz≤y. Denote by δ(P) the supremum of the cardinals κ such thatPcontains a subset of pairwise disjoint elements with cardinal number κ. P. Erdös and A. Tarski (Ann. of Math.44, 1943, 315–329) proved that, unless δ(P) is weakly inaccessible,Pcontains a subset of pairwise disjoint elements with cardinal number δ(P). J. Dauns and L. Fuchs (J. Algebra115, 1988, 297–302) defined theGoldie dimensionof a moduleM, denoted by GdM, as the supremum of all cardinals κ such thatMcontains the direct sum of κ nonzero submodules. They proved that, unless GdMis weakly inaccessible,Mcontains a direct sum of GdMsubmodules. In this paper, a unified proof of these two results is given. It is also shown that similar results hold in the context of modular lattices and abelian categories.

Varieties of Anticommutativen-ary Algebras

A fundamental problem in the theory ofn-ary algebras is to determine the correct generalization of the Jacobi identity. This paper describes some computational results on this problem using representations of the symmetric group. It is well known that over a field of characteristic 0 any variety ofn-ary algebras can be defined by multilinear identities. In the anticommutative case, it is shown that forn≤8 the[formula]-dimensionalS2n−1-module of multilinear identities in which each term involves twon-ary products (i.e., two pairs ofn-ary anticommutative brackets) decomposes as the direct sum of thendistinct simple modules labelled by thenpartitions of 2n−1 in which only 1 and 2 occur as parts. In the casesn=3 (resp.n=4), the kernel of the commutator expansion map and a generator for each of the 7 (resp. 15) nonzero submodules are determined. The paper concludes with some conjectures forn≥5.