artinian ring
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2022 ◽  
Vol 29 (01) ◽  
pp. 167-180
Author(s):  
Mahdi Reza Khorsandi ◽  
Seyed Reza Musawi

Let [Formula: see text] be a commutative ring and [Formula: see text] the multiplicative group of unit elements of [Formula: see text]. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, [Formula: see text], corresponding to a multiplicative subgroup [Formula: see text] of [Formula: see text] and a nonempty subset [Formula: see text] of [Formula: see text] with [Formula: see text], as the graph with vertex set [Formula: see text]and two distinct vertices [Formula: see text] and [Formula: see text] being adjacent if and only if there exists [Formula: see text] such that [Formula: see text]. In this paper, we characterize all Artinian rings [Formula: see text] for which [Formula: see text] is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring [Formula: see text] for which [Formula: see text] has finite nonorientable genus, [Formula: see text] must be a finite ring. Finally, it is proved that for a given positive integer [Formula: see text], the number of finite rings [Formula: see text] for which [Formula: see text] has nonorientable genus [Formula: see text] is finite.


Author(s):  
Zongyang Xie ◽  
Zhongkui Liu ◽  
Xiaoyan Yang

Let [Formula: see text] be a commutative artinian ring and [Formula: see text] a small Ext-finite Krull–Schmidt [Formula: see text]-abelian [Formula: see text]-category with enough projectives and injectives. We introduce two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text] in terms of the representable functors from the stable category of [Formula: see text] to category of finitely generated [Formula: see text]-modules. Moreover, we define two additive functors [Formula: see text] and [Formula: see text], which are mutually quasi-inverse equivalences between the stable categories of this two full subcategories. We give an equivalent characterization on the existence of [Formula: see text]-Auslander–Reiten sequences using determined morphisms.


Author(s):  
Bülent Saraç

Two obvious classes of quasi-injective modules are those of semisimples and injectives. In this paper, we study rings with no quasi-injective modules other than semisimples and injectives. We prove that such rings fall into three classes of rings, namely, (i) QI-rings, (ii) rings with no middle class, or (iii) rings that decompose into a direct product of a semisimple Artinian ring and a strongly prime ring. Thus, we restrict our attention to only strongly prime rings and consider hereditary Noetherian prime rings to shed some light on this mysterious case. In particular, we prove that among these rings, QIS-rings which are not of type (i) or (ii) above are precisely those hereditary Noetherian prime rings which are idealizer rings from non-simple QI-overrings.


Author(s):  
Nil Orhan Ertaş ◽  
Rachid Tribak

We prove that a ring [Formula: see text] has a module [Formula: see text] whose domain of projectivity consists of only some injective modules if and only if [Formula: see text] is a right noetherian right [Formula: see text]-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor [Formula: see text]-modules is closed under direct summands if and only if [Formula: see text] is a right Bass ring. A ring [Formula: see text] is said to have no right max-p-middle class if every right [Formula: see text]-module is either projective or max-poor. It is shown that if a commutative noetherian ring [Formula: see text] has no right max-p-middle class, then [Formula: see text] is the ring direct sum of a semisimple ring [Formula: see text] and a ring [Formula: see text] which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field [Formula: see text] of [Formula: see text] has a proper [Formula: see text]-submodule which is not complete in its [Formula: see text]-topology. Then we show that a commutative noetherian hereditary ring [Formula: see text] has no right max-p-middle class if and only if [Formula: see text] is a semisimple ring.


Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Mehdi Khoramdel ◽  
Saboura Dolati Pish Hesari

We introduce the notion of semi-poor modules and consider the possibility that all modules are either injective or semi-poor. This notion gives a generalization of poor modules that have minimal injectivity domain. A module [Formula: see text] is called semi-poor if whenever it is [Formula: see text]-injective and [Formula: see text], then the module [Formula: see text] has nonzero socle. In this paper the properties of semi-poor modules are investigated and are used to characterize various families of rings. We introduce the rings over which every module is either semi-poor or injective and call such condition property [Formula: see text]. The structure of the rings that have the property [Formula: see text] is completely determined. Also, we give some characterizations of rings with the property [Formula: see text] in the language of the lattice of hereditary pretorsion classes over a given ring. It is proved that a ring [Formula: see text] has the property [Formula: see text] iff either [Formula: see text] is right semi-Artinian or [Formula: see text] where [Formula: see text] is a semisimple Artinian ring and [Formula: see text] is right strongly prime and a right [Formula: see text]-ring with zero right socle.


2021 ◽  
Vol 45 (01) ◽  
pp. 63-73
Author(s):  
S. M. SAADAT MIRGHADIM ◽  
M. J. NIKMEHR ◽  
R. NIKANDISH

Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by AR, is a graph whose vertex set is the set of all non-zero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) ∩ Ann(J) = (0). In this paper, we characterize all Artinian rings for which both of the graphs AR and AR (the complement of AR), are chordal. Moreover, all Artinian rings whose AR (and thus AR) is perfect are characterized.


2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Bikash Barman ◽  
Kukil Kalpa Rajkhowa

PurposeThe authors study the interdisciplinary relation between graph and algebraic structure ring defining a new graph, namely “non-essential sum graph”. The nonessential sum graph, denoted by NES(R), of a commutative ring R with unity is an undirected graph whose vertex set is the collection of all nonessential ideals of R and any two vertices are adjacent if and only if their sum is also a nonessential ideal of R.Design/methodology/approachThe method is theoretical.FindingsThe authors obtain some properties of NES(R) related with connectedness, diameter, girth, completeness, cut vertex, r-partition and regular character. The clique number, independence number and domination number of NES(R) are also found.Originality/valueThe paper is original.


2021 ◽  
Vol 7 (4) ◽  
pp. 5106-5116
Author(s):  
Yousef Alkhamees ◽  
◽  
Sami Alabiad
Keyword(s):  

<abstract><p>An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.</p></abstract>


Author(s):  
Le Van Thuyet ◽  
Phan Dan ◽  
Truong Cong Quynh

In this paper, by taking the class of all [Formula: see text] (or [Formula: see text]) right [Formula: see text]-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes) and a right [Formula: see text]-ring (or a right noetherian [Formula: see text]-ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes). By using isosimple-projective preenvelope, we obtained that if [Formula: see text] is a semiperfect right FGF ring (or left coherent ring), then every isosimple right [Formula: see text]-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.


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