Rings characterized by their right ideals or cyclic modules

arXiv (Cornell University), 2022

In the present paper, modules over integral domains and principal ideal domains that are proper essential extensions of some submodules are classified. We introduce a new class of modules that we call SM modules and show that the class of Artinian modules, locally finite modules, and modules of finite lengths are all proper subclasses of SM modules. We also show that non-semisimple SM modules possess essential socles. Further, we show that non-semieimple modules over integral domains with nonempty torsion-free parts do not possess essential socles.

Journal of pure and applied algebra, 1992

Colloquium Mathematicum, 2015

A module M over a ring R satises the restricted minimum condition (RMC) if M=N is Artinian for every essential submodule N of M. A ring R satises right RMC if R R satises RMC as a right module. It is proved that (1) a right semiartinian ring R satises right RMC if and only if R=Soc(R) is Artinian, (2) if a semilocal ring R satises right RMC and Soc(R) = 0, then R is Noetherian if and only if the socle length of E(R=J(R)) is at most !, and (3) a commutative ring R satises RMC if and only if R=Soc(R) is Noetherian and every singular module is semiartinian.

Journal of Pure and Applied Algebra, 2004

A module M is said to satisfy the condition (˝ *) if M is a direct sum of a projective module and a quasi-continuous module. By Huynh and Rizvi (J. Algebra 223 (2000) 133; Characterizing rings by a direct decomposition property of their modules, preprint 2002) rings over which every countably generated right module satisÿes (˝ *) are exactly those rings over which every right module is a direct sum of a projective module and a quasi-injective module. These rings are called right˝ *-semisimple rings. Right˝ *-semisimple rings are right artinian. However, in general, a right˝ *-semisimple rings need not be left˝ *-semisimple. In this note, we will prove a ring-direct decomposition theorem for right and left˝ *-semisimple rings. Moreover, we will describe the structure of each direct summand in the obtained decomposition of these rings.

Arxiv preprint math/0407270, 2004

Abstract. Let R be a commutative Noetherian ring. It is shown that R is Artinian if and only if every R-module is good, if and only if every R-module is representable. As a result, it follows that every nonzero submodule of any representable R-module is representable if and only if R is ...

Journal of Algebra, 2011

In a recent paper, Alahmadi, Alkan and López-Permouth defined a module M to be poor if M is injective relative only to semisimple modules, and a ring to have no right middle class if every right module is poor or injective. We prove that every ring has a poor module, and characterize rings with semisimple poor modules. Next, a ring with no right middle class is proved to be the ring direct sum of a semisimple Artinian ring and a ring T which is either zero or of one of the following types: (i) Morita equivalent to a right PCI-domain, (ii) an indecomposable right SI-ring which is either right Artinian or a right V-ring, and such that soc(T T) is homogeneous and essential in T T and T has a unique simple singular right module, or (iii) an indecomposable right Artinian ring with homogeneous right socle coinciding with the Jacobson radical and the right singular ideal, and with unique non-injective simple right module. In case (iii) either T T is poor or T is a QF-ring with J (T) 2 = 0. Converses of these cases are discussed. It is shown, in particular, that a QF-ring R with J (R) 2 = 0 and homogeneous right socle has no middle class.

Publicacions Matemàtiques, 2013

We carry out a study of rings R for which Hom R (M, N) = 0 for all nonzero N ≤ M R. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Köthe ring R is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when R is retractable.

Communications in Algebra

By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them virtually semisimple modules. A module R M is called completely virtually semisimple if each submodules of M is a virtually semisimple module. A ring R is then called left (completely) virtually semisimple if R R is a left (compleatly) virtually semisimple R-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring R is left completely virtually semisimple if and only if R ∼ = k i=1 M ni (D i) where k, n 1 , ..., n k ∈ N and each D i is a principal left ideal domain. Moreover, the integers k, n 1 , ..., n k and the principal left ideal domains D 1 , ..., D k are uniquely determined (up to isomorphism) by R.

Kyungpook Mathematical Journal

We say a module a semicommutative module if for any and any , implies . This paper gives various properties of reduced, Armendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and be a p.p.-module, then is a semicommutative module iff is an Armendariz module. For any ring R, R is semicommutative iff A(R, ) is semicommutative. Let R be a reduced ring, it is shown that for number and , is semicommutative ring but is not.

Journal of Algebra, 1985

Journal of the Australian Mathematical Society, 1980

Let R be a ring in which the multiplicative semigroup is completely semisimplc. If R has the maximum (respectively, minimum) condition on principal multiplicative ideals, then R is semipnme artinian (respectively, a direct sum of dense rings of finite-rank linear transformations of vector spaces over division rings).

2016

Abstract. We say a module MR a semicommutative module if for any m ∈ M and any a ∈ R, ma = 0 implies mRa = 0. This paper gives various properties of reduced, Ar-mendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and MR be a p.p.-module, then MR is a semicommutative module iff MR is an Armendariz module. For any ring R, R is semicommutative iff A(R,α) is semicommu-tative. Let R be a reduced ring, it is shown that for number n ≥ 4 and k = [n/2], T kn (R) is semicommutative ring but T k−1n (R) is not. 1.

Journal of Algebra, 1990

This paper continues the study of an R-module M through properties of the category a[M] of submodules of M-generated modules. It is shown that a self-projective M is locally artinian if and only if every cyclic module in a[M] is a direct sum of an M-injective and a finitely cogenerated module. From this we derive that a ring T with local units is locally left artinian if and only if every module in T-Mod is a direct sum of a T-injective module and a finitely cogenerated module. For rings with unity this was proved in Huynh--Dung [3) applying Osofsky's observation about left-injective regular rings R: For an infinite set of orthogonal idempotents {eA}, the factor module R/C,, Re, is not injective. Combined with a categorical equivalence Huynh-Dung's result is also used in our own proof.

Let 𝑀 be an 𝑅-semimodule and 𝑁 non-zero subsemimodule of 𝑀. We say that 𝑁 is an essential subsemimodule of 𝑀, if 𝑁 ∩ 𝐾 ≠ (0) for every nonzero subsemimodule 𝐾 of 𝑀. In this paper we study some useful results on essential subsemimodules and singular semimodule of semiring.

Journal of Mathematical Sciences, 2009

All right R-modules are I0-modules if and only if either R is a right SV-ring or R/I (2) (R) is an Artinian serial ring such that the square of the Jacobson radical of R/I (2) (R) is equal to zero. All rings are assumed to be associative and with nonzero identity element. Expressions such as "an Artinian ring" mean that the corresponding right and left conditions hold. A submodule X of the module M is said to be superfluous in M if X + Y = M for every proper submodule P of the module M. Following [9], we call a module M an I 0-module if every nonsuperfluous submodule of M contains a nonzero direct summand of the module M. It is clear that I 0-modules are weakly regular modules, considered in [1-3, 8]; a module M is said to be weakly regular if every submodule of M that is not contained in the Jacobson radical of M contains a nonzero direct summand of M. Weakly regular modules are studied in [1-3; 6; 8; 9; 11, Chap. 3; 12-14], and other papers. A ring R is called a right generalized SV-ring if every right R-module is weakly regular. It is clear that A is a right generalized SV-ring provided each right A-module is an I 0-module. In addition, it follows from the presented paper that the Jacobson radical of any right module over a right generalized SV-ring is superfluous; therefore, every right module over a right generalized SV-ring is an I 0-module. The aim of the paper is the study of generalized right SV-rings. The main result of the present paper is Theorem 1.