Singular Submodule Research Articles

Z-Small Submodules and Z-Hollow Modules

A submodule Ϝ of an R-module Ε is called small in Ε if whenever , for some submodule W of Ε , implies . In this paper , we introduce the notion of Ζ-small submodule , where a proper submodule Ϝ of an R-module Ε is said to be Ζ-small in Ε if , such that , then , where is the second singular submodule of Ε . We give some properties of Ζ-small submodules . Moreover , by using this concept , we generalize the notions of hollow modules , supplement submodules, and supplemented modules into Ζ-hollow modules, Ζ-supplement submodules, and Ζ-supplemented modules. We study these concepts and provide some of their relations .

On subprojectivity domains of g-semiartinian modules

The aim of this paper is to reveal the relationship between the proper class generated projectively by g-semiartinian modules and the subprojectivity domains of g-semiartinian modules. A module [Formula: see text] is called g-semiartinian if every nonzero homomorphic image of [Formula: see text] has a singular simple submodule. It is proven that every g-semiartinian right [Formula: see text]-module has an epic projective envelope if and only if [Formula: see text] is a right PS ring if and only if every subprojectivity domain of any g-semiartinian right [Formula: see text]-module is closed under submodules. A g-semiartinian module whose domain of subprojectivity as small as possible is called gsap-indigent. We investigated the structure of rings whose (simple, coatomic) g-semiartinian right modules are gsap-indigent or projective. Furthermore, over right PS rings, necessary and sufficient condition to be gsap-indigent module was determined.

Modules which are invariant under t-automorphisms of their injective hulls

ABSTRACTLet R be an arbitrary ring with identity and M, a right R-module with Z2(M) the second singular submodule of M. We call an endomrphism φ of M, a t-automorphism if defined by is an isomorphism and we call M, a t-automorphism-invariant module if it is invariant under t-automorphisms of its injective hulls. This paper is devoted to investigation of various properties and characterizations of t-automorphism-invariant modules. We show that a module M is t-automorphism invariant if and only if every t-isomorphism between two t-essential submodules of M extends to an endomorphism of M.

Total graph of a module with respect to singular submodule

Let R be a commutative ring with unity and M be an R-module. We introduce the total graph of a module M with respect to singular submodule Z(M) of M as an undirected graph T(Γ(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x+y∈Z(M). We investigate some properties of the total graph T(Γ(M)) and its induced subgraphs Z(Γ(M)) and Z¯(Γ(M)). In some aspects, we have noticed some sort of finiteness.

SP-INJECTIVITY OF MODULES AND RINGS

Let >M and N be two R-modules. NR is called singular M-p-injective if for every singular M-cyclic submodule X of MR, every homomorphism from X to N can be extended to a homomorphism from M to N. MR is quasi-singular prinicipally injective if M is a singular M-p-injective module. It is shown that a ring R is right non-singular if and only if every right R-module is singular R-p-injective if and only if factors of singular R-p-injective modules are singular R-p-injective. A singular R-module M is injective if and only if M is N-sp-injective for every R-module N. Finally, we characterize quasi-sp-injective modules in terms of their endomorphism rings.

A generalization of lifting modules

In this paper we introduce the notions of G 1L-module and G 2L-module which are two proper generalizations of -lifting modules. We give some character- izations and properties of these modules. We show that a G 2 L-module decomposes into a semisimple submodule M1 and a submodule M2 of M such that every non-zero submodule of M2 contains a non-zero -cosingular submodule. Throughout this article, all rings are associative with an identity, and all modules are unitary right R modules. Let M be an R-module. By NM(N � � M) we mean that N is a submodule (direct summand) of M. A submodule N of a module M is called essential in M if for every nonzero submodule L of M, N L 0 (denoted by Ne M) and A submodule N of a module M is called small in M if for every proper submodule L of M, N +L M (denoted by N ≪ M). A module M is called hollow if every proper submodule of M is small in M. M is called a small module if there exists a module T such that M ≪ T. If N=K ≪ M=K, then K is called a cosmall submodule of N in M. A submodule N of M is called coclosed if N has no proper cosmall submodule. Recall that the singular submodule Z(M) of a module M is the set of m 2 M with mI = 0 for some essential ideal I of R. If Z(M) = M (Z(M) = 0), then M is called a singular (non-singular) module. Let K, N be submodules of M. Following (14), as a

Schwach-injektive Moduln

An ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> R$-module $M$ is called weakly injective if for every extension $M \subset X$, $M$ is coclosed in $X$. We show for every noetherian, local, one-dimensional integral domain $R$ with field of fractions $K$ and completion $\hat{R}$, that $\hat{R} \bigotimes\limits_R K$ as $\hat{R}$-module as well as $K/R$ as $R$-module are weakly injective. Moreover, we show that $K/R'$ is weakly injective iff $R$ is analytically unramified. For certain module classes over an arbitrary noetherian ring $R$ the weak injectivity can be described by means of the singular submodule $Z(M)$ as well as by the dual submodule $\overline{Z}(M)=\bigcap\,\{U \subset M\mid M/U$ is small in its injective hull$\}$ (see Talebi and Vanaja, \emph{Commun.\ Algebra} 30 (2002), 1449--1460 and Z\"oschinger, \emph{Commun.\ Algebra} 33 (2005), 3389--3404). If $R$ is local and if $M$ possesses a primary decomposition we prove: every factor module of $M$ is weakly injective iff for every ${\mathfrak{p}} \in \operatorname{Coass}(M)$ the ring of fractions $R_{\mathfrak{p}}$ is a field and the fibre ring $\hat{R} \bigotimes\limits_R \kappa({\mathfrak{p}})$ is semisimple.

ON INJECTIVITY AND P-INJECTIVITY

The following results are extended from P-injective rings to AP-injective rings: (1) R is left self-injective regular if and only if R is a right (resp. left) AP-injective ring such that for every flnitely generated left R-module M, R(M=Z(M)) is projec- tive, where Z(M) is the left singular submodule of RM; (2) if R is a left nonsingular left AP-injective ring such that every maximal left ideal of R is either injective or a two-sided ideal of R, then R is either left self-injective regular or strongly regular. In addition, we answer a question of Roger Yue Chi Ming (13) in the positive. Let R be a ring whose every simple singular left R-module is Y J-injective. If R is a right MI-ring whose every essential right ideal is an essen- tial left ideal, then R is a left and right self-injective regular, left and right V -ring of bounded index.

A remark on a theorem of Y. Kurata

In [K] Y.Kurata proved that the Goldie torsion theory splits centrally for dual rings. Here we extend his result to semilocal rings with left essential socle such that Soc (RR) ⊆ Soc (RR). An example will demonstrate that our observation extends Kurata’s result. All rings are associative ring with unit and all R-modules are unital. The singular submodule of a left R-module M is denoted by Z(RM). We abbreviate S := Soc (RR) and J := Jac (R) for the left socle resp. the Jacobson radical. We denote the left Goldie torsion theory, that is the hereditary torsion theory generated by all singular left R-modules, by τG and the torsion submodule of a module M by τG(M). τG is said to be centrally splitting if τG(R) is a ring direct summand of R. A classical result of Allin and Dickson [AD] states that τG is centrally splitting for a ring R if and only if R is a direct product of a semisimple ring and a ring with essential left singular ideal. Lemma 1. Let R be a ring with essential left socle S. Then (1) S = S ⊕ (S ∩ Z(RR)), where S is projective and R/S is τG-torsion. (2) J is τG-torsion if and only if S J = 0. Proof. The socle can be decomposed as S = S0 ⊕ S1 where S1 := S ∩ Z(RR) and S0 is a projective left R-module. S ⊆ S0, because for x, y ∈ S with x = x0+x1 and y = y0+y1 where x0, y0 ∈ S0 and x1, y1 ∈ S1. The product xy = xy0 ∈ S0 as xy1 ∈ SZ(RR) = 0. Thus S ⊆ S0 holds and there exists a left module S such that S0 = S ⊕ S. We have SS ⊆ S∩ S = 0. If RS is essential in RR, then S becomes singular (as it is annihilated by S) and must be zero as it is also projective. Thus S0 = S . Also R/S becomes τG-torsion as S1 ' S/S and R/S are 1991 Mathematics Subject Classification. 16S90.

Principally quasi-injective modules

An R-module M is called principally quasi-injective if each R-hornomorphism from a principal submodule of M to M can be extended to an endomorphism of M. Many properties of principally injective rings and quasi-injective modules are extended to these modules. As one application, we show that, for a finite-dimensional quasi-injective module M in which every maximal uniform submodule is fully invariant, there is a bijection between the set of indecomposable summands of M and the maximal left ideals of the endomorphism ring of M Throughout this paper all rings R are associative with unity, and all modules are unital. We denote the Jacobson radical, the socle and the singular submodule of a module M by J(M), soc(M) and Z(M), respectively, and we write J(M) = J. The notation N ⊆ess M means that N is an essential submodule of M.

A note on continuous rings

Throughout this note we consider associative rings with identity and unitary modules. For a module M over a ring R we write MR (resp. aM) to indicate that M is a right (resp. left) R-module. By Soc (MR), Z (MR) and d (MR) we denote the socle, the singular submodule and the Jacobson radical of M R. respectively. A ring R is called semiprimary if the Jacobson radical d of R is nilpotent and the factor ring R/d is semisimple Artinian. A module M is called continuous if (i) every submodule of M is contained essentially in a direct summand of M. and (ii) every submodule of M which is isomorphic to a direct summand of M, is itself a direct summand of M. A ring R is right continuous if R R is continuous. For a detailed study on continuous rings and modules, we refer to [5].

P-injectivity of simple pre-torsion modules

V-rings and their generalisations have been studied extensively in recent years [2], [3], [5],[6], [7]. All the rings we consider will be associative rings with 1 ≠ 0 and all the modules considered will be unitary left R-modules. All the concepts will be left-sided unless otherwise mentioned. Thus by an ideal in R we mean a left ideal of R. A ring R is called a V-ring (respectively a GV-ring) if every simple (resp. simple, singular) module is injective. An R-module M is called p-injective if any homomorphism f: I → M with I a principal left ideal of R can be extended to a homomorphism g: R → M. A ring R is called a p-V-ring (resp. a p-V'-ring) if every simple (resp. simple, singular) module over R is p-injective. The object of the present paper is to introduce torsion theoretic generalizations of p-V-rings and prove results similar to those obtained by Yue Chi Ming about p-V-rings and p-V'-rings [6], [7]. For any M ∈ R-mod, J(M) will denote the Jacobson radical of M and Z(M) the singular submodule of M. For any λ ∈ R, we denote the left annihilator { r ∈ R| rλ =0 } of λ in R by l(λ).

Rings all of whose torsion quasi-injective modules are injective

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.

Semiperfect FPF Rings

In this paper we derive some of the structure of semiperfect FPF rings. A ring is right FPF if every f.g. faithful right module is a generator. For semiperfect right and left FPF rings we show that if all one sided zero divisors are two sided zero divisors, then the classical and maximal quotient rings coincide (all four of them) and are self-injective. We show that if the intersection of the powers of the Jacobson radical is zero, then right and left regular elements are regular. Also, we show right FPF semiperfect rings contain the singular submodule of their injective hulls and that every finitely generated module contained in the injective hull and containing the ring is isomorphic to the ring.

Semiperfect FPF rings

In this paper we derive some of the structure of semiperfect FPF rings. A ring is right FPF if every f.g. faithful right module is a generator. For semiperfect right and left FPF rings we show that if all one sided zero divisors are two sided zero divisors, then the classical and maximal quotient rings coincide (all four of them) and are self-injective. We show that if the intersection of the powers of the Jacobson radical is zero, then right and left regular elements are regular. Also, we show right FPF semiperfect rings contain the singular submodule of their injective hulls and that every finitely generated module contained in the injective hull and containing the ring is isomorphic to the ring.

$V$-rings relative to hereditary torsion theories

of modules closed under cyclicsubmodules, homomorphic images and extensions. Applying Theorem 2 for the Goldie and the Lambek torsion theories, we obtain Corollaries5 and 6. We considerin Corollary 5 a ring R (calleda right V(G)-ring) for which every singular simple right R-module isinjective,and in Corollary 6 a right V(L)-ring for which every dense right idealis an intersection of maximal right ideals. We characterize V-rings in terms of V(G)rings or V(L)-rings in Proposition 8 which is closelyrelatedto Theorem 8 in [7]. In Theorem 9 itis proved that commutative V(G)-rings turn out to be V-rings. In this connection two examples are given to show that neither commutative V(L)-rings nor V(G)-rings are V-rings. Throughout this paper R is a ring with a unit,every right R-module is unitaland Mod-j? is the category of right i?-modules. For a right i?-module M, Z(M), E{M) and J{M) denote the singular submodule of M, the injective hullof M and theintersection of allmaximal submodules of M. A right Rmodule M is called2-injectivefor a subclass 2of Mod-i? if Homfi(―, M) preserves the exactness for every exact sequence of right i?-modules Q-*A^B―>C ―0 with Ce2＼

Triangular Representations of Splitting Rings

The term “splitting ring” refers to a nonsingular ring R such that for any right R-module M, the singular submodule of M is a direct summand of M. If R has zero socle, then R is shown to be isomorphic to a formal triangular matrix ring $\left ( {\begin {array}{*{20}{c}} A & 0 \\ B & C \\ \end {array} } \right )$, where A is a semiprime ring, C is a left and right artinian ring, and $_C{B_A}$ is a bimodule. Also, necessary and sufficient conditions are found for such a formal triangular matrix ring to be a splitting ring.

Bicommutators of nice injectives

Given any injective module I R , its bicommutator S contains as a subring the ring Q of quotients of R relative to the largest torsion theory for which I R is torsionfree. Q is even a dense subring of S if I R is sufficiently “nice”: every torsionfree factor module of Q R is divisible. Then S is the completion of Q in the finite topology, which here coincides with a certain I-adic topology, generalizing the usual P-adic topology on finitely generated modules over Noetherian local rings. Examples of nice injectives are: all injectives when R is right hereditary, injectives with zero singular submodule and the injective hull of R/ P when R is commutative and P is a prime ideal. It will be noticed that, in a number of concepts depending on the injective module I R , explicit reference to the latter has been suppressed. Thus we write “torsion” rather than “ I-torsion”, but we have retained “ I-adic” to contrast with “ P-adic”. The present paper supplements [8], but some care has been taken to make it self-contained.

A splitting ring of global dimension two

In this paper an example is given of a ring with left global dimension 2 having the property that the singular submodule of any R-module A is a direct summand of A. Although the example given is quite specific, the methods can be used to construct a fairly large class of these rings.

Quotient rings of endomorphism rings of modules with zero singular submodule

Throughout this paper (R, M, N, S) will denote a Morita context satisfying a certain nonsingularity condition. For such contexts we give necessary and sufficient conditions in terms of M and R for S to have a semisimple maximal left quotient ring; respectively a full linear maximal left quotient ring, a semisimple classical left quotient ring. In doing so we extend the corresponding well-known theorems for rings (employing them in the process) to endomorphism rings.