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

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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.

All rings are assumed to be associative and with nonzero identity element, and all modules are assumed to be unitary. A ring is said to be right (resp., left) invariant if all right (resp., left) ideals of it are ideals. Expressions such as “an invariant ring” mean that the corresponding right and left conditions hold. A right module M over a ring A is called a multiplication module if for every submodule N of M , there exists an ideal B of the ring A such that N = MB. A module MA is said to be flat if for every left A-module L, the natural group homomorphism M ⊗A L is a monomorphism. If all submodules of any right or left module over a ring A are flat, then one says that the weak global dimension of the ring A does not exceed 1; in this case, we write w.gl.dim(A) ≤ 1 for brevity. In [7], it is proved that every faithful multiplication module over a commutative ring is a flat module. If A is a commutative ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive semiprime ring [5]. If A is an invariant semiprime ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive ring [13, Theorem 7.31]. The main result of the present paper is Theorem 1.

Let R be a commutative ring with identity and M be unitary (left) R-module. The principal aim of this paper is to study the relationships between relatively cancellation module and multiplication modules, pure submodules and Noetherian (Artinian) modules.

INTRODUCTION In the broadest sense, this is a study of commutative rings which satisfy the (finitely) pseudo-Froben[us (or (F)PF) condition: All (finitely generated) faithful modules generate the category mod-R of all R-modules. These rings include: Pr[[fer rings, almost maximal valuation rings, self-injective rings, e.g. , quasi-Frobenius (QF) and pseudo-Frobenius (PF) rings, and finite products of these. (In fact, any product of commutative FPF rings is FPF [34]; hence, any product of commutative PF rings is FPF (cf. §9).) If R is FPF, so is its (classical)ring of quotients Qc~(R) and its maximal quotient ring Qmax(R). All known FPF rings are (classically) quotientinjective in the sense that Qcf is injective. 2 We conjecture that all FIDF rings are quot[ent-injective, and prove this in the three cases: (i) local rings (Proposition 7 and Theorem 9B): (2) Noether[an rings (Theorem ii; Endo' s Theorem [25J~ (3) reduced rings (Proposition 3B and Theorem 4). Moreover, any FPF commutative ring R splits, R = R 1 X R2, where R 1 is semihered[tary, and R 2 has essential nilradical. (If R is semilocal or Noetherian, then R 2 is injective. ) Thus any reduced FPF ring has regular [njective Qc~' and conversely any quotient-injective sem[hered[tary ring is FPF (Theorem 4). A ring iS pre-(l~PF iff all(finitely generated)faithful ideals are generators, and we

Let $V$ be a submodule of a direct sum of some elements in $\mathcal{U}$, and $X$ be a submodule of a direct sum of some elements in $\mathcal{N}$, where $\mathcal{U}$ and $\mathcal{N}$ are families of $R$-modules. A $\mathcal{U}$-free module is a generalization of a free module. According to the definition of $\mathcal{U}$-free module, we define three kinds of projective$_{\mathcal{U}}$ in this research, i.e., projective$_{\underline{\mathcal{U}}}$, projective$_{\mathcal{U}}$ module, and strictly projective$_{\mathcal{U}}$ module. The notion of strictly projective$_{\mathcal{U}}$ is a generalization of the projective module. In this research, we discuss the relationship between projective modules and the three types of modules. Furthermore, we show that the properties of $\mathcal{U}$ impact the properties of the projective$_{\mathcal{U}}$ module so that we can determine some properties of the projective$_{\mathcal{U}}$ module based on the properties of the family of $\mathcal{U}$ of $R$-modules.

Totally acyclic complexes

Let M be a nonsingular quadratic module over a commutative ring R. The primary goal of this chapter is the proof of the following fact: If M is free of finite rank, then C(M) has a special element, or, equivalently, the Arf algebra A(M) of M is isomorphic to R[X](X2 - aX - b) for some a and b in R with a2 + 4b ∈ R*. This has the important consequence that the results developed in Chapters 7 and 8 for C(M) hold for any free M. In addition, we will establish that the discriminant a2 + 4b of X2 - aX - b is the classical discriminant of the quadratic module M (except for a factor 2 if rank M is odd). It was established in Chapter 9D that the Arf algebra A(M) is a separable quadratic algebra. The structural features of this algebra will play a decisive role in the proofs. Throughout, R is an arbitrary commutative ring.

Let A be a commutative ring and let E be an A-module with a finite free resolution (see below for definition). Extending results known previously for noetherian rings, it is shown that ann ( E ) = annihilator {\text {ann}}(E)=\text {annihilator} of E is trivial if and only if the Euler characteristic of E = χ ( E ) > 0 E = \chi (E) > 0 ; in addition, if χ ( E ) = 0 , ann ( E ) \chi (E) = 0,{\text {ann}}(E) is dense (i.e. ann ( ann ( E ) ) = 0 {\text {ann}}({\text {ann}}(E)) = 0 ). Also, a local ring is constructed with its maximal ideal with a finite free resolution but consisting exclusively of zero-divisors and thus, contrary to the noetherian case, one does not always have a nonzero divisor in ann ( E ) {\text {ann}}(E) if χ ( E ) = 0 \chi (E) = 0 . Finally, if E has a finite resolution by (f.g.) projective modules it turns out that ann ( ann ( E ) ) {\text {ann}}({\text {ann}}(E)) is generated by an idempotent element.

We introduce and investigate the notion of $\gc$-projective modules over (possibly non-noetherian) commutative rings, where $C$ is a semidualizing module. This extends Holm and J{\o}rgensen's notion of $C$-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite $\gc$-projective dimension, showing in particular that they admit $\gc$-projective approximations, a generalization of the maximal Cohen-Macaulay approximations of Auslander and Buchweitz. Over a local (noetherian) ring, we provide necessary and sufficient conditions for a $G_C$-approximation to be minimal.

In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including [Formula: see text]-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings [Formula: see text] for which every faithful multiplication [Formula: see text]-module is endo-reduced is precisely that of reduced rings; while the class of rings [Formula: see text] for which every multiplication [Formula: see text]-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PR-modules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module is PR-module if and only if for every , is pure submodule of (or for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules.

Introduction. In '[12] Thrall introdcuced three generalizations of the quasiFrobenius (=QF) algebras of Nakayama [9], [10]. In this paper we shall be concerned with ring theoretic generalizations of two of these Thrall algebras-namely QF-2 algebras and QF-3 algebras. If R is a ring then a one-sided ideal of R is primitive in case it is generated by a primitive idempotent, and an R-module is minimalfaithful in case it is faithful and has no proper faithful direct summand. Extending Thrall's original definitions to (two-sided) artinian rings we have: QF-2 rings: An artinian ring is QF-2 in case each of its primitive one-sided ideals has a simple socle. QF-3 rings: An artinian ring is QF-3 in case it has (to within isomorphism) a unique minimal faithful module. It is not difficult to show that QF rings are both QF-2 and QF-3 (see [2, ??58-59]). Moreover, Thrall [12] has shown that QF-2 algebras are QF-3 but not necessarily QF. Most of the information about QF-2 and QF-3 rings is limited to finite dimensional algebras (see [8], [12], [13]). Two notable exceptions generalize to QF-3 rings results known to hold for QF-3 algebras almost from their inception. Specifically, Jans [7] has characterized QF-3 rings as those artinian rings whose injective hulls are projective and Harada [5] has shown that the QF-3 property is actually two-sided (i.e., a QF-3 ring has a unique minimal faithful right module). In ?2 of this paper we obtain ideal theoretic characterizations of the injective projective modules (and hence of the unique minimal faithful module) over left QF-3 rings. Our main results appear in ?3. With the aid of Morita's duality theorems [8] we obtain characterizations of QF-3 rings that are analogous to Nakayama's original definition of QF rings in terms of socles of primitive one-sided ideals [10], his characterization of QF-rings in terms of the double annihilator property for one-sided ideals [10], and the fact (see [8, ?14]) that QF rings are precisely those artinian rings for which the functor HomR ( , R) provides a duality between the categories of finitely generated and finitely generated right R-modules.

Abstract. In this paper, we give some properties on submoduletransforms. 0. IntroductionLet M be a module over commutative ring Rwith identity, Stheset of nonzero divisors of Rand R S the total quotient ring of R. For anonzero ideal Iof R, let I 1 = fx2R S jxIRg. Iis said to be aninvertible ideal of Rif II 1 = R. Put T= ft2Sjtm= 0 for some m2Mimplies m= 0g. Then Tis a multiplicatively closed subset of Sand ifMis torsion free, then T= S([9,Proposition 1.1]). In particular, if Misa faithful multiplication module then Mis torsion free ([4,Lemma 4.1])and so T= S. So in this case, R T = R S . Let Nbe a submodule of M. Ifx= rt 2R T and n2N, then we say that xn2Mif there exists m2Msuch that tm= rn. Then this is a well de ned operation([9,p399]). Fora submodule N of M, N 1 = fx2R T jxN Mg= [M : R T N]. Wesay that Nis invertible in Mif NN 1 = Mand Mis called a Dedekind(resp. Pruf er) module providing that every nonzero (resp. every nonzero nitely generated) submodule of Mis invertible.Mis called a multiplication module if every submodule Nof Mhasthe form IM for some ideal I of R. An R module M is said to befaithful if Ann (M) = [0 :

In this article, some new characterizations of Gorenstein projective, injective, and flat modules over commutative noetherian local rings are given.