Modules whose injectivity domains are restricted to semi-artinian modules

2021 ◽  
pp. 2250150
Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Mehdi Khoramdel ◽  
Saboura Dolati Pish Hesari
Keyword(s):  
Artinian Ring ◽  
Artinian Modules

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.

Generalized Co-Cohen–Macaulay and Co-Buchsbaum Modules

2007 ◽  
Vol 14 (02) ◽  
pp. 265-278
Author(s):  
Nguyen Tu Cuong ◽  
Nguyen Thi Dung ◽  
Le Thanh Nhan
Keyword(s):  
Artinian Modules ◽  
Basic Properties ◽  
Local Homology

We study two classes of Artinian modules called co-Buchsbaum modules and generalized co-Cohen–Macaulay modules. Some basic properties and characterizations of these modules in terms of 𝔮-weak co-sequences, co-standard sequences, multiplicity, local homology modules are presented.

A special property in Artinian modules

2017 ◽  
Vol 11 ◽  
pp. 31-34
Author(s):  
Mansooreh Maani-Shirazi
Keyword(s):  
Special Property ◽  
Artinian Modules

Artinian Modules Over a Matrix Ring

2000 ◽  
pp. 101-105 ◽  
Author(s):  
K. I. Pimenov ◽  
A. V. Yakovlev
Keyword(s):  
Matrix Ring ◽  
Artinian Modules

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

Homological Aspects of Semigroup Gradings on Rings and Algebras

1999 ◽  
Vol 51 (3) ◽  
pp. 488-505 ◽  
Author(s):  
W. D. Burgess ◽  
Manuel Saorín
Keyword(s):  
Euler Characteristic ◽  
Division Ring ◽  
Artinian Ring ◽  
Global Dimension ◽  
Monomial Algebra ◽  
Finite Dimensional ◽  
Cartan Matrices ◽  
Torsion Free ◽  
Rings And Algebras ◽  
Finite Dimensional Algebras

AbstractThis article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup Σ. Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the Σ-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties.A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert Σ-series in the associated path incidence ring.The rationality of the Σ-Euler characteristic, the Hilbert Σ-series and the Poincaré-Betti Σ-series is studied when Σ is torsion-free commutative and A is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

A GENERALIZATION OF BAER'S LOWER NILRADICAL FOR MODULES

2007 ◽  
Vol 06 (02) ◽  
pp. 337-353 ◽  
Author(s):  
MAHMOOD BEHBOODI
Keyword(s):  
Prime Ring ◽  
Artinian Ring ◽  
Prime Radical ◽  
Nilpotent Elements ◽  
Noetherian Domain ◽  
Prime Submodules ◽  
Strongly Nilpotent ◽  
Commutative Domain ◽  
Torsion Modules

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals [Formula: see text] and for all submodules N ⊆ M, [Formula: see text] implies that [Formula: see text] or [Formula: see text]. We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.rad R(M)] and Baer's lower nilradical for a module [denoted by Nil *(RM)]. For a module RM, cl.rad R(M) is defined to be the intersection of all classical prime submodules of M and Nil *(RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.rad R(M) = Nil *(RM) and, for any module M over a left Artinian ring R, cl.rad R(M) = Nil *(RM) = Rad (M) = Jac (R)M. In particular, if R is a commutative Noetherian domain with dim (R) ≤ 1, then for any module M, we have cl.rad R(M) = Nil *(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim (R) ≤ 1 (or over a commutative domain R with dim (R) ≤ 1), every semiprime submodule of any module is an intersection of classical prime submodules.

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

2014 ◽  
Vol 21 (02) ◽  
pp. 249-256 ◽  
Author(s):  
G. Aalipour ◽  
S. Akbari ◽  
M. Behboodi ◽  
R. Nikandish ◽  
M. J. Nikmehr ◽  
...  
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Vertex Set

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

Characterization Theorems of S-Artinian Modules

2021 ◽  
Author(s):  
Mehmet Özen ◽  
OsamaA. Naji ◽  
Unsal Tekir ◽  
Kar Ping Shum
Keyword(s):  
Artinian Modules ◽  
Characterization Theorems

Example of an Injective Module which is Not Nice

1974 ◽  
Vol 17 (1) ◽  
pp. 133-134
Author(s):  
Gerhard O. Michler
Keyword(s):  
Artinian Ring ◽  
Injective Module ◽  
Finite Topology ◽  
Factor Module ◽  
Ring Of Quotients

In [1] Lambek calls the injective R-module I nice if every torsionfree factor module of the ring of quotients Q of R with respect to lis divisible. If lis nice then g is a dense subring of the bicommutator BicRI of I with respect to the finite topology (see [1, Proposition 2]). We now give an example of an injective R-module over an Artinian ring R which is not nice. Since R is Artinian, Q=BicRI, by Proposition B of [1].Before we give the example, we state the following, which depends on [2] for terminology.

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